Here are two cool functions defined by power series:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\sum_{n=1}^{\infty}\frac{z^{n-1}}{(1-z^{n})(1-z^{n+1})}=\left\{\begin{array}{cc}\frac{1}{(1-z)^2},&\mbox{ if }

|z|<1 \\\frac{1}{z(1-z)^2}, & \mbox{ if } |z|>1\end{array}\right.[/tex]

and

[tex]\sum_{n=1}^{\infty}\frac{z^{2^{n-1}}}{1-z^{2^{n}}}=\left\{\begin{array}{cc}\frac{z}{1-z},&\mbox{ if }

|z|<1 \\\frac{1}{1-z}, & \mbox{ if } |z|>1\end{array}\right.[/tex]

The first sum is from (pg. 59, #1) A Course of Modern Analysis by E.T. Whittaker & G.N. Watson, and the second sum (I checked this one rigorously, but not the first) is from (pg. 267, #100b) Theory and Applications of Infinite Series by K. Knopp.

So, any other functions defined by power series that converge to one function for |z|< r and to another function for |z|>r ?

A discussion of the analytic continuation of functions (and, perhaps, the natural boundaries thereof) is nearly expected--and somewhat encouraged. But please, post more nifty sums like these.

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# Anyone know any more sums like these?

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