About

Complex geometric series

In analogy to the real series the elements of the complex exponential sequence follow the rule

z_n= aⁿ

z_n i the nth member of the sequence, with index n an integer, including 0. The parameter a is a complex number, as is z. With z₀ =1

the members are: 1, a, a², a³, a⁴..... z_n = aⁿ

The complex geometric partial sum series S_n is formed by consecutive addition of the members of the sequence:

S_n = Σ_oⁿ a ^m with 0 ≤ m ≤ n ; S_n = 1 + a + a ²....+ aⁿ

This simulation calculates 500 elements of the sequence. By drawing the red point in the left chart, a is defined. The blue points are the members of the sequences, as they are of the partial sum series in the right chart.

The case of the real geometric series is observed when a is a point on the real axis.

The complex geometric series converges if the absolute value of a is smaller than 1, a lying inside of the red unit circle. If convergent, the limit is

lim_n→∞ S_n = lim_n→∞ Σ_oⁿ a^m = 1/(1-a)

It is at the center of the small green circle in the right chart of the series.

The marked point at the unit circle is the first term of both sequence and series, the real number 1. The prominent second point in the sequence is a, which can be drawn with the mouse.

When a has an imaginary component and abs(a)<1 sequence and series spiral towards the convergence point (limit). When small imaginary parts are chosen the whole series will be on one Riemann sheet. For larger imaginary parts the spiral can have many revolutions, the points of which cover more than one Riemann sheet. The spiraling is much more pronouncedly visible than with the exponential series: because of the slower convergence many calculated points are distinguishable with the geometric series.

When a approaches the unit circle from the inside complex patterns may be observed as the series converges. When the real part is negative, the spiral splits into several arms (the real one series splits in two). An interesting split is observed at a real part of zero and a close to i. When a crosses the unit circle the series diverges.

For angles of the a vector of 2π/N, with N integer, the spirals have N- fold symmetry. This is best seen when a is close to the unit circle.

The series increases very fast as a approaches the unit circle. The scales are self adjusting and the unit circle may appear just like a point.

For abs(a) > 0 the series diverges. The unit circle becomes invisible and the series diverges in spirals to infinity. For better viewing draw the diagram up to full screen size.

Near the inner rim of the unit circle convergence may be so slow that 500 calculated points are not sufficient to approach the series limit at the center of the green circle.

Complex exponential series

The elements of the complex exponential sequence follow the rule

z_n+1= z_n*a / n

(for comparison:  geometric series :  z_n+1= z_n*a )

z_n i the nth member of the sequence, with index n a positive entity, including 0. The growth parameter a is a complex number. With

z₀= 1

the members are: 1, a /1 , a ²/(1*2), a³/(1*2*3), a⁴/(1*2*3*4)....

z_n =aⁿ/ n!

( n! = 1*2*3*4*...*n n! = n -faculty)

The complex exponential partial sum series S_n is formed by consecutive addition of the members of the sequence:

S_n= Σ₀ⁿ a^m/m! 0 ≤ m ≤n ; S_n = 1 + a + a ²/2....+a ⁿ/n!

This simulation calculates 500 elements of the sequence. By drawing the red point in the left chart, a is defined. The blue points are the members of the sequence. In the right chart they are those of the partial sum series.

The case of the real exponential series is observed when a is a point on the real axis.

Members of the exponential sequence always converge to zero. Its partial sum series converges to a finite number for all finite a.

lim (S_n= Σ₀ⁿ a^m/m!) = e^a; e = 2.71828....Euler number

When a has an imaginary component the series spirals toward the convergence point (limit). Its sourrounding is marked by a small green circle.

For small imaginary parts all points will be on a single Riemann sheet. For large imaginary parts of a one observes multiple revolutions of the spiral, corresponding to points in different Riemann sheets.  The effect is less obvious than with the complex geometric series because of the fast convergence of the exponential, for which most calculated points lie within the small green circle.

E1: Reset. With a ={0.5, 0} the partial sum series converges to 2.

Shift a along the real axis and compare results with the simulation of the real geometric series.

E2: Choose a close to {1, 0}

The condition of convergence of the partial sum series is obviousely abs(a ) < 1.

What is the condition for the elements of the sequence?

E3: Choose a around {-1, 0}. Observe both charts.

E4:  Choose abs(a) < 1 with a small imaginary part and watch the behavior of the series.

E5: Increase the imaginary part and reflect what determines the character of the sequence spiral:

multiplication by a for each member increases the angle to the x-axes by the angle of a: arctg [imaginarypart(a)/realpart(a)]. Soon the points will be on multiple Riemann sheets.

E6: Look for a, where the spirals form straight radial arms, and analyze the cause of these symmetries.

E1:  Reset. With a ={1,0} the partial sum series converges to e = 2.718....

Shift a along the real axis and compare results with the simulation of the geometric series.

E2: Choose real(a) ≈ 2.5, with arbitrary imaginary part.

What happens with the sequence?

Why does the series converge, while the absolute value of its sequence increases for the first members?

E3: Compare the character of convergence to that of the complex geometric series. What is responsible for the unlimited convergence of the exponential?

E4: Shift a along the imaginary axis, starting at 0.

How does the limit change? What is its formula? r = const = 1 = 1 * (cosa + i sina) ➙ e^iy = cosy + i siny

E5: Shift a parallel to the imaginary axis.

How does the limit change? What is its formula?

r = const = e^a = e^real(a) e^im(a) = e^real(a) (cos(im(a) + i sin(im(a)) ➙ e^{x + iy} = e^x(cosy + i siny ) Express this in words.

this file was created by Dieter Roess in August 2009

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

 

Translations

Code	Language		Translator	Run

Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee

Overview:

This briefing document summarizes the key information presented on the Open Educational Resources / Open Source Physics @ Singapore website regarding their JavaScript simulation applet for visualizing complex geometric and exponential series. The resource provides interactive HTML5 simulations to aid in learning and teaching mathematics, specifically focusing on complex numbers and series convergence.

Main Themes and Important Ideas:

The primary focus of the resource is to illustrate and explain the behavior of complex geometric and exponential series through an interactive simulation. Key concepts covered include:

1. Complex Geometric Series:

• Definition: The complex geometric series is defined as the sum of a sequence where each term is the previous term multiplied by a constant complex number 'a'. The nth term is given by zn = a^n, starting with z0 = 1.
• Partial Sum: The partial sum Sn is the sum of the first n+1 terms: Sn = 1 + a + a^2 + ... + a^n = Σ(m=0 to n) a^m. The simulation calculates the first 500 elements.
• Convergence: The crucial condition for the convergence of a complex geometric series is that the absolute value (modulus) of the common ratio 'a' must be less than 1 (abs(a) < 1). This is visually represented by a red unit circle in the simulation.
• "The complex geometric series converges if the absolute value of a is smaller than 1, a lying inside of the red unit circle."
• Limit of Convergence: If the series converges, its limit as n approaches infinity is given by the formula: lim(n→∞)Sn = 1/(1-a). This limit is depicted by a small green circle in the simulation's right chart.
• Visualization: The simulation provides two charts:
• Left Chart: Shows the individual terms of the sequence (blue points) as determined by the complex number 'a' (red point, draggable by the user).
• Right Chart: Shows the partial sums of the series (blue points).
• Behavior Based on 'a':Real 'a': When 'a' lies on the real axis, the simulation mirrors the behavior of a real geometric series.
• Imaginary Component (|a| < 1): If 'a' has an imaginary component and its absolute value is less than 1, both the sequence and the partial sum series spiral towards the convergence point. Larger imaginary parts can lead to spirals covering multiple Riemann sheets.
• "When a has an imaginary component and abs(a)<1 sequence and series spiral towards the convergence point (limit)."
• 'a' Approaching the Unit Circle: As 'a' gets closer to the unit circle from the inside, complex convergence patterns emerge. A negative real part can cause the spiral to split into arms. At a real part of zero and 'a' near i, an interesting split is observed.
• 'a' Crossing the Unit Circle: If the absolute value of 'a' is greater than or equal to 1 (abs(a) ≥ 1), the series diverges. For abs(a) > 0, the series spirals to infinity.
• Symmetry: For angles of the 'a' vector equal to 2π/N (where N is an integer), the spirals exhibit N-fold symmetry, especially noticeable when 'a' is close to the unit circle.
• Slow Convergence: Near the inner rim of the unit circle, convergence can be slow, requiring more than 500 terms to approach the limit.

2. Complex Exponential Series:

• Definition: The complex exponential sequence follows the rule zn+1 = zn * a / n, with z0 = 1. The nth term is given by zn = a^n / n!.
• Partial Sum: The partial sum Sn is given by Sn = Σ(m=0 to n) a^m / m! = 1 + a + a^2/2! + ... + a^n / n!. The simulation also calculates 500 elements for this series.
• Convergence: The members of the exponential sequence always converge to zero, and its partial sum series converges to a finite number for all finite complex values of 'a'.
• "Members of the exponential sequence always converge to zero. Its partial sum series converges to a finite number for all finite a."
• Limit of Convergence: The limit of the exponential partial sum series is lim(n→∞)Sn = e^a, where 'e' is Euler's number (approximately 2.71828). The limit is marked by a small green circle in the right chart.
• Visualization: Similar to the geometric series, the left chart shows the sequence terms, and the right chart displays the partial sums.
• Behavior Based on 'a':Imaginary Component: When 'a' has an imaginary component, the series spirals towards its convergence point. Larger imaginary parts can lead to multiple revolutions on different Riemann sheets, though this effect is less pronounced than in the geometric series due to faster convergence.
• Faster Convergence: The exponential series converges much faster than the geometric series, meaning most calculated points tend to cluster closer to the limit.

3. Interactive Experiments:

The resource suggests several experiments for users to explore the behavior of both series by manipulating the value of 'a':

• Geometric Series Experiments:Observing convergence for real 'a' within and outside the unit interval (-1, 1).
• Examining the behavior when 'a' is close to {1, 0} and {-1, 0}.
• Investigating the effect of small and increasing imaginary parts on the spiral.
• Identifying values of 'a' that result in straight radial arms and analyzing the cause.
• Exponential Series Experiments:Observing convergence to 'e' when a = {1, 0}.
• Analyzing the sequence and series behavior for real(a) ≈ 2.5 with an arbitrary imaginary part, noting that the sequence can initially increase while the series still converges.
• Comparing the convergence characteristics to the geometric series, highlighting the "unlimited convergence" of the exponential series.
• Shifting 'a' along the imaginary axis and parallel to it to observe how the limit changes and relating it to the formula e^(iy) = cosy + i siny and e^(x+iy) = e^x (cosy + i siny).

4. Pedagogical Value:

The resource is explicitly designed for learning and teaching mathematics using simulations. It aims to provide a visual and interactive way for students to understand abstract concepts like complex numbers, geometric and exponential series, and the conditions for their convergence.

5. Technical Information:

• The simulation is an HTML5 JavaScript applet, making it embeddable in webpages via an iframe.
• It is part of a larger collection of simulations for mathematics and physics.
• The underlying EasyJavaScriptSimulations Library may have separate licensing terms for commercial use.

Conclusion:

This JavaScript simulation applet provides a valuable tool for visualizing and understanding complex geometric and exponential series. Its interactive nature allows users to explore the impact of the complex parameter 'a' on the behavior and convergence of these series, reinforcing key mathematical concepts through visual experimentation. The resource is a component of a broader initiative to leverage simulations in open educational resources for mathematics and physics education.

 

Study Guide: Complex Geometric and Exponential Series

Quiz

1. What is the fundamental rule that defines the elements of a complex geometric sequence?
2. Explain the condition for convergence of a complex geometric series and state the formula for its limit when it converges.
3. How does the behavior of a complex geometric series change when the absolute value of the parameter 'a' crosses the unit circle?
4. Describe the visual representation of a converging complex geometric series in the provided simulation, focusing on the movement of the partial sums.
5. What is the key difference in the recursive rule defining the elements of a complex exponential sequence compared to a complex geometric sequence?
6. Do the individual members of a complex exponential sequence converge, and if so, to what value? What about the partial sum series?
7. How does the convergence of a complex exponential series visually differ from that of a complex geometric series in the simulation?
8. What happens to the spiral of a complex geometric series when the angle of the vector 'a' is a specific fraction of 2π (i.e., 2π/N, where N is an integer)?
9. In the context of the complex exponential series, explain why the series converges for all finite values of the growth parameter 'a', even when the absolute value of the sequence terms initially increases.
10. How does shifting the complex parameter 'a' along the imaginary axis affect the limit of the complex exponential partial sum series, and what is the formula for this limit?

Quiz Answer Key

1. The elements of a complex geometric sequence follow the rule zn = a^n, where 'a' is a complex number and 'n' is an integer index starting from 0. The first term, z₀, is equal to 1.
2. A complex geometric series converges if the absolute value (modulus) of the complex number 'a' is smaller than 1 (i.e., |a| < 1), meaning 'a' lies inside the unit circle in the complex plane. If convergent, the limit of the series is given by the formula 1/(1-a).
3. When the absolute value of 'a' crosses the unit circle from the inside (|a| becomes greater than 1), the complex geometric series diverges. Instead of spiraling towards a finite limit, the partial sums spiral outwards to infinity.
4. In the simulation, a converging complex geometric series is visualized by blue points (representing the partial sums) in the right chart that spiral towards a central green circle, which denotes the limit of the series. The coloring of the points changes from red to cyan as more terms are added, illustrating the progression of the partial sums.
5. The key difference in the recursive rule is the inclusion of the index 'n' in the denominator for the exponential series: zn+1 = zn * (a / n). In contrast, the geometric series follows zn+1 = zn * a.
6. Yes, the individual members of the complex exponential sequence always converge to zero for any finite complex number 'a'. The complex exponential partial sum series converges to a finite number for all finite values of 'a', specifically to e^a, where 'e' is Euler's number.
7. The convergence of the complex exponential series is generally faster than that of the geometric series. Visually, in the simulation, the spiral of the exponential series tends to have most of its calculated points clustered within the small green circle around the convergence point, making the spiraling effect less pronounced than in the geometric series.
8. For angles of the vector 'a' equal to 2π/N (where N is an integer), the spirals of the complex geometric series exhibit N-fold symmetry. This symmetrical pattern is most clearly observable when 'a' is located close to the unit circle.
9. While the absolute values of the initial terms of the exponential sequence can increase if |a| > 1, the presence of the n! (n-factorial) term in the denominator of the general term (a^n / n!) causes the terms to eventually decrease rapidly and converge to zero. This rapid decrease ensures the convergence of the partial sum series.
10. Shifting 'a' along the imaginary axis, starting at 0, changes the limit of the complex exponential partial sum series according to the formula e^(iy) = cos(y) + i sin(y), where 'y' is the imaginary part of 'a'. This means the limit lies on the unit circle in the complex plane, with its angle determined by the imaginary component of 'a'.

Essay Format Questions

1. Compare and contrast the definitions, conditions for convergence, and visual representations of complex geometric and complex exponential series as presented in the simulation. Discuss the factors that contribute to their differing behaviors.
2. Analyze the role of the parameter 'a' (a complex number) in determining the convergence and the geometric properties (e.g., spiraling, symmetry) of the complex geometric series. How does the simulation illustrate these relationships?
3. Discuss the concept of Riemann sheets in the context of complex geometric and exponential series with large imaginary components in their parameter 'a'. Explain why spiraling can lead to points residing on multiple Riemann sheets and how this is suggested by the simulation.
4. Evaluate the educational value of the JavaScript simulation applet in understanding the behavior of complex geometric and exponential series. How do the interactive features and visual representations aid in grasping abstract mathematical concepts?
5. Explore the connection between real geometric and exponential series and their complex counterparts, as described in the source material. How do the properties of real series serve as an analogy for understanding the more general complex cases, and what new behaviors emerge in the complex domain?

Glossary of Key Terms

• Complex Number: A number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit defined as the square root of -1.
• Geometric Series: A series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
• Complex Geometric Series: A geometric series where the common ratio is a complex number.
• Exponential Series: A series whose terms involve exponentiation, often related to the exponential function. In this context, it refers to the series Σ (a^m / m!).
• Partial Sum Series (Sn): The sum of the first 'n' terms of a sequence or series. For a geometric series, Sn = 1 + a + a² + ... + a^n. For the exponential series, Sn = 1 + a + a²/2! + ... + a^n/n!.
• Convergence: The property of a series where its partial sums approach a finite limit as the number of terms increases.
• Divergence: The property of a series where its partial sums do not approach a finite limit as the number of terms increases.
• Absolute Value (Modulus) of a Complex Number: The distance of a complex number from the origin in the complex plane. For z = a + bi, |z| = √(a² + b²).
• Unit Circle: A circle in the complex plane centered at the origin with a radius of 1.
• Riemann Sheet: One of the infinitely many copies of the complex plane which together form the Riemann surface of a multi-valued complex function. Moving onto a different Riemann sheet corresponds to going around a branch point of the function.
• Growth Parameter (a): The complex number that determines the behavior of the terms in both the geometric (as the common ratio) and exponential series.
• Factorial (n!): The product of all positive integers up to a given positive integer 'n' (e.g., 5! = 1 × 2 × 3 × 4 × 5). 0! is defined as 1.
• Euler's Number (e): A mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm.
• N-fold Symmetry: A geometric figure has N-fold rotational symmetry if it looks the same after being rotated by 360°/N degrees around its center.

Sample Learning Goals

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For Teachers

The geometric series is written as:
,
where a is the first term of the sequence, and r is a constant real number called the common ratio.

The series converges to a finite value if |r|<1, and the limit that the series converges to in that case is given by 1/(1-a).

This definition still holds for when the common ratio is a complex number z, with the only difference being that instead of having the absolute value of r be less than 1 for the series to converge, the modulus of z has to be less than 1. This is shown in the simulation as the unit circle. As long as the common ratio (denoted a in the simulation) remains inside the unit circle, the series will converge to a finite value. Otherwise, it diverges.

There are two graphs in the simulation. The leftmost graph shows the terms of the series, the colour turning from red to cyan as the terms progress. The rightmost graph shows the partial sums up to each term in the series, and the green square denotes the limit of the series. The colouring of the points in the rightmost graph is the same as the leftmost graph, to make the mapping clearer.

The first term in the series is fixed at z = 1.

There is also an exponential function, where the terms in the series also converge, but they do so at a much slower rate than the geometric series.

Research

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Video

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 Version:

1. http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
2. http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

Other Resources

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Complex Geometric Series

1. What is a complex geometric series and how is it defined? A complex geometric series is analogous to a real geometric series, but its terms and common ratio can be complex numbers. It begins with a first term (which is 1 in this simulation) and each subsequent term is obtained by multiplying the previous term by a constant complex number, denoted as 'a' in the simulation. The nth term of the sequence is given by zn = a^n, starting with z0 = 1. The partial sum Sn is the sum of the first n+1 terms: Sn = 1 + a + a^2 + ... + a^n.

2. Under what condition does a complex geometric series converge, and what is its limit? A complex geometric series converges if the absolute value (or modulus) of the common ratio 'a' is less than 1 (|a| < 1). In the simulation, this is represented by 'a' lying inside the red unit circle in the left chart. If the series converges, its limit as n approaches infinity is given by the formula: lim(n→∞) Sn = 1 / (1 - a). This limit is visualized as the center of the small green circle in the right chart.

3. How does the imaginary component of the common ratio 'a' affect the behavior of the complex geometric series when it converges (|a| < 1)? When the common ratio 'a' has an imaginary component and its absolute value is less than 1, both the sequence of terms and the partial sum series spiral towards the convergence point in the complex plane. The size of the imaginary part influences the number of revolutions in the spiral; larger imaginary parts can lead to the spiral covering multiple Riemann sheets. The spiraling effect is more visible in the geometric series compared to the exponential series due to its slower convergence.

4. What happens to the complex geometric series when the absolute value of the common ratio 'a' is greater than or equal to 1 (|a| ≥ 1)? When the absolute value of 'a' is greater than or equal to 1, the complex geometric series diverges. If |a| > 1, the terms of the sequence grow in magnitude, and the partial sums spiral outwards to infinity. If |a| = 1 (i.e., 'a' lies on the unit circle), the series generally also diverges, and complex patterns may be observed as 'a' approaches the unit circle from the inside.

5. What are some interesting patterns observed in the complex geometric series simulation when 'a' is close to the unit circle or has specific angular properties? When 'a' approaches the unit circle from the inside, complex convergence patterns can be observed. If the real part of 'a' is negative, the spiral of the partial sums may split into several arms. A notable split occurs when the real part is zero and 'a' is close to the imaginary unit 'i'. For angles of the 'a' vector equal to 2π/N (where N is an integer), the spirals exhibit N-fold symmetry, which is most apparent when 'a' is near the unit circle.

Complex Exponential Series

6. How does the definition and behavior of the complex exponential series differ from the complex geometric series? The complex exponential series has terms defined by zn = a^n / n!, where 'a' is a complex growth parameter. Unlike the geometric series where each term is a constant multiple of the previous one, in the exponential series, each term is the previous term multiplied by a/n. While the terms of the geometric sequence converge to zero only if |a| < 1, the members of the exponential sequence always converge to zero for any finite 'a'. The partial sum series of the exponential series converges to e^a for all finite complex numbers 'a', whereas the geometric series converges only if |a| < 1. The convergence of the exponential series is generally faster, making the spiraling towards the limit less pronounced.

7. What determines the limit of the complex exponential partial sum series? The limit of the complex exponential partial sum series as n approaches infinity is given by e^a, where 'a' is the complex growth parameter. If 'a' has a real part (x) and an imaginary part (y), then e^a = e^(x + iy) = e^x * (cos(y) + i sin(y)). This means the magnitude of the limit is determined by the real part of 'a' (e^x), and the angle (in the complex plane) is determined by the imaginary part of 'a' (y). Shifting 'a' along the imaginary axis changes the angle of the limit while keeping its magnitude constant. Shifting 'a' parallel to the imaginary axis changes the angle of the limit, and the magnitude of the limit is determined by the constant real part of 'a'.

8. Why does the complex exponential series always converge to a finite value for any finite 'a', even if the absolute values of its initial terms increase? The convergence of the complex exponential series is guaranteed by the factorial term (n!) in the denominator of its general term (a^n / n!). While for small values of n, the magnitude of a^n might increase (especially if |a| > 1), the factorial grows much faster than any exponential function. As n becomes large, the n! term dominates, causing the terms a^n / n! to rapidly approach zero. Since the terms of the sequence eventually become very small and sum to a finite value, the partial sum series converges for all finite complex values of 'a'. This is in contrast to the geometric series, where the terms only approach zero if |a| < 1.