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Closed Form for an Infinite Series

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Find a closed form for the infinite series 1+ x + x^2 + x^3 +.... show that the closed form is only valid if /x/< 1, where x may be a complex number (i.e. x=x1 +ix2)

In the complex plane, draw a diagram that shows successive values of 1, 1+x, 1+x+x^2 , 1+x+x^2+x^3, etc. for x=i. Does the series converge for x=i?

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This series is a geometric series and so its infinite sum is 1/(1-x) and this is its closed form.

The ...

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Application of Bolzano-Weierstrass Theorem

Recall that a point z0 is an accumulation point of a set S if each deleted neighborhood of z0 contains at least one point in S.

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One form of the Bolzano-Weierstrass theorem can be stated as follows:

An infinite set of points lying in a closed bounded region R has at least one accumulation point in R.

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Theorem 1:

Given a function f and a point z0, suppose that:
a) f is analytic at z0
b) f(z0) = 0 but f(z) is not identically equal to zero in any neighborhood of z0.

Then f(z) does not equal 0 throughout some deleted neighborhood 0<|z - z0|< epsilon of z0.

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Using the Bolzano-Weierstrass Theorem and Theorem 1, prove that if a function f is analytic in a region R consisting of all points inside and on a simple closed contour C (except possibly poles inside C) and if all the zeros of f in R are interior to C and are of finite order, then those zeros must be finite in number.

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