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

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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