Question: "How can I prove this result?∑k=0∞2k22k+1=1"

lovesickgirlsrd5

Answered question

2022-02-24

How can I prove this result?

\sum_{k = 0}^{\infty} \frac{2^{k}}{2^{2^{k}} + 1} = 1

Chance Mill

Beginner2022-02-25Added 8 answers

Make it into a telescopic sum as follows

\sum_{k = 0}^{n} \frac{2^{k}}{2^{2^{k}} + 1} = \sum_{k = 0}^{n} \frac{2^{k} (2^{2^{k}} - 1)}{(2^{2^{k + 1}} - 1)}

= \sum_{k = 0}^{n} \frac{2^{k} (2^{2^{k}} - 1)}{(2^{2^{k + 1}} - 1)}

= \sum_{k = 0}^{n} \frac{2^{k} (2^{2^{k}} + 1)}{(2^{2^{k + 1}} - 1} - \frac{2^{k + 1}}{(2^{2^{k + 1}} - 1)}}

= 1 - \frac{2^{n + 1}}{(2^{2^{n + 1}} - 1)} \to 1

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