By computing the sum of all Bernoulli numbers via Borel summation (I learned this technique from Varadarajan's excellent book Euler through time. A new look at old themes, 2006) I found that $\sum $\sum B_n = \int_0^\infty \frac{t}{e^{2t}-e^t} dt$ dt$$ and discovered numerically that this expression equals $\zeta(2)-1$. The web is not very good for finding out where this can be found in print. Where should I look, and how can equations such as $\zeta(2) $\zeta(2) = 1 + \int_0^\infty \frac{t}{e^{2t}-e^t} dt$ frac{t}{e^{2t}-e^t}\ dt$$ be proved?
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Integral expression for zeta(2)By computing the sum of all Bernoulli numbers via Borel summation (I learned this technique from Varadarajan's excellent book Euler through time. A new look at old themes, 2006) I found that $\sum B_n = \int_0^\infty \frac{t}{e^{2t}-e^t} dt$ and discovered numerically that this expression equals $\zeta(2)-1$. The web is not very good for finding out where this can be found in print. Where should I look, and how can equations such as $\zeta(2) = 1 + \int_0^\infty \frac{t}{e^{2t}-e^t} dt$ be proved?
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