lovesickgirlsrd5
2022-02-24
Answered

How can I prove this result?

$\sum _{k=0}^{\mathrm{\infty}}\frac{{2}^{k}}{{2}^{{2}^{k}}+1}=1$

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Chance Mill

Answered 2022-02-25
Author has **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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$\sum _{n=1}^{\mathrm{\infty}}\frac{2n}{{3}^{n+1}}$

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Using green's theorem, find the counterclockwise circulation and outward flux for field F and curve C, if:

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I looked through other questions and didn't quite find an answer to this. I know that the intermediate value theorem says that $\text{Im}((f-g)(t))$ and $\text{Re}((f-g)(t))$ must be 0 at some point, but I'm not sure how to show that this must occur at the same point.

I looked through other questions and didn't quite find an answer to this. I know that the intermediate value theorem says that $\text{Im}((f-g)(t))$ and $\text{Re}((f-g)(t))$ must be 0 at some point, but I'm not sure how to show that this must occur at the same point.

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$\sum _{n=0}^{\mathrm{\infty}}\frac{\mathrm{cos}\left(n\pi \right)}{3n!+1}$

$\sum _{n=0}^{\mathrm{\infty}}\frac{\mathrm{cos}\left(n\pi \right)}{3n!+1}$

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Make a substitution in the above taylore series to get the first 7 terms for the Taylor Series

$f(x)=\frac{1}{\sqrt{1-x}}$

Make a substitution in the above taylore series to get the first 7 terms for the Taylor Series

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