5
$\begingroup$

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?

1 Answers 1

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} $

  • 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