I have a question about this limit. Calculate$\lim_{t\to{1-0}}(1-t)(\frac{t}{1+t}+\frac{t^2}{1+t^2}+...+\frac{t^n}{1+t^n}+...)$ Can anyone help?
Evaluate the sum $\lim_{t\to{1-0}}(1-t)(\frac{t}{1+t}+\frac{t^2}{1+t^2}+...+\frac{t^n}{1+t^n}+...)$
5
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calculus
limits
1 Answers
4
Let $h=-\log t$. Then
$ \begin{eqnarray} \lim_{t\nearrow 1}(1-t)\sum_{k=1}^\infty\frac{t^k}{1+t^k} &=& \lim_{t\nearrow 1}\sum_{k=1}^\infty(1-t)\frac1{1+t^{-k}} \\ &=& \lim_{h\searrow 0}\sum_{k=1}^\infty(1-\mathrm e^{-h})\frac1{1+\mathrm e^{kh}} \\ &=& \lim_{h\searrow 0}\sum_{k=1}^\infty(h+O(h^2))\frac1{1+\mathrm e^{kh}} \\ &=& \int_0^\infty\frac1{1+\mathrm e^x}\mathrm dx \\ &=& \log2\;. \end{eqnarray} $
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0@Peter: I see, interesting. I guess those are different situations in that the summation limit occurs in the summand in one case but not the other. – 2012-04-21