Question

Nice derivation of $\sum_{n=1}^\infty \frac{1}{n} \left( \frac{q^{2n}}{1-q^n}+\frac{\bar q^{2n}}{1-\bar q^n}\right)=-\sum_{m=2}^\infty \ln |1-q^m|^2$

score 104 · Answer 1 · 2026-04-16 18:04:42

Nice derivation of $\sum_{n=1}^\infty \frac{1}{n} \left( \frac{q^{2n}}{1-q^n}+\frac{\bar q^{2n}}{1-\bar q^n}\right)=-\sum_{m=2}^\infty \ln |1-q^m|^2$

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147

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109

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score 46 · Answer 5 · 2026-04-14 02:31:14

46

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875

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136

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score 290 · Answer 10 · 2026-04-14 02:38:12

290

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List Question

Nice derivation of $\sum_{n=1}^\infty \frac{1}{n} \left( \frac{q^{2n}}{1-q^n}+\frac{\bar q^{2n}}{1-\bar q^n}\right)=-\sum_{m=2}^\infty \ln |1-q^m|^2$

Compute the following sum for any x?

Summation Proof

Closed Form for Factorial Sum

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$?

Binary representation of the real numbers

Question on simplification of $\sum_{n=1}^{\infty}\frac{2}{(2n+1)(2n+3)}$?

Is the series: $\,\sum_{n=1}^{\infty}\frac{\mathrm{e}^{-in}}{n}\,$ divergent?

Intuition to why average of the square of a positive integer and the integer itself is the sum of all numbers from 1 to the integer?

Closed-form of $\sum_{k=0}^{\infty} \frac{k^a\,b^k}{k!}$

a complicated Sum

Convergence of $\sum_{n=1}^{\infty} \frac{Sin^2(n)}{n}$

Solving an unusual equation

How to make a sum vanish?

Proof of Nesbitt's Inequality: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$?

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