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I needed to give an example of a power series that satisfies the following conditions: interval of convergence is [-1,1] and is conditionally convergence at both -1 and 1.

Is it even possible to have both endpoints conditionally converge without using a "piece-wise series"? I got stuck and this all I've gotten so far:

I know that the series should be $\sum_{n}^{\infty} C_nx^{n}$ since the interval of convergence is [-1,1] which means it's centered at 0 and I know the $\lim_{n\to\infty} \frac{C_{n+1}}{Cn} = 1$ by the ratio test.

However, the conditonally convergent thing at both endpoints is messing me up. It makes it hard to find a $C_n$. Is making the series piece-wise the only way? I.e., include a $(-1)^n$ when x = 1 to make it alternating and not have $(-1)^n$ when x = -1?

Any tips or is piece wise the way to go?

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One can use $$\sum_{n=1}^\infty (-1)^n \frac{x^{2n}}{n}$$

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  • $\begingroup$ Thanks!! I was stuck with $$\sum_{n}^{\infty} \frac{(-1)^n}{n} x^n$$ $\endgroup$ – Chubbles Apr 5 '16 at 20:04
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    $\begingroup$ You are welcome. Squaring gets rid of minus sign issues. $\endgroup$ – André Nicolas Apr 5 '16 at 20:10

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