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Consider the below power series:

$\sum\limits_{n=1}^\infty \dfrac{x^{n}}{n^{2}}$

I know that it converges for $x\in [-1,1]$ and the sum $s(x)$ of the series is given by:

$s(x) = - \int\limits_{0}^{x} \frac{\ln(1-t)}{t} dt$ for all $x\in ]-1,1[$

I now have to explain why the integrand can be extended uniquely to be a contionuos function on $]-1,1[$, even though it is only defined in $]-1,0[\cup ]0,1[$.

I'm completely lost on this question. What do they want me to do? I can see from the series that $s(0) = 0$ but what is the statement "extended uniquely" supposes to mean?

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    Sorry I forgot to multiply it with -12012-05-25

0 Answers 0