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Find the value of

$$\frac{1}{3+1}+\frac{2}{3^2+1}+\frac{4}{3^4+1}+\frac{8}{3^8+1}+\cdots+\frac{2^{2006}}{3^{2^{2006}}+1}$$

I try to form the summation into telescoping sum but I stuck.

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1 Answer 1

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More generally, define $S_n = \frac{1}{3+1}+\frac{2}{3^2+1}+\frac{4}{3^4+1}+\frac{8}{3^8+1}+\cdots+\frac{2^{n}}{3^{2^{n}}+1}$.

Hint: $\frac{2^{n}}{3^{2^n}+1}=\frac{2^{n}}{3^{2^n}-1}-\frac{2^{n+1}}{3^{2^{n+1}}-1}$.

Another way is to use induction.

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