I really need help with this exercise. Consider the power series(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\sum_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{2n+1}.[/tex]

for [tex]z\in\mathbb{C}.[/tex]

I need to answer the following questions:

a)Is the series convergent for z = 1?

This is easy; just plug in z = 1 and observe that the alternating series obtained is convergent using some basic theorems and stuff.

b)Is the series convergent for z = i?

Here I'm in trouble; the absolute series (series of absolute values) diverges, but that tells me nothing... Any hints?

c)Show that the radius of convergence is R=1.

I have done this, but in a complicated way that isn't the right way, for sure. What's confusing me is that this is not a power series in the standard form, [tex]\sum a_n z^n[/tex] - if you write this series in this way every second term is 0 (corresponding to even n's), so the standard formulas in my book for finding radius of convergence are not applicable (at least, I'm not able to apply them).

Thanks.

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# Power series

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