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Find sum of the series:

$\sum\limits_{n=0}^{\infty}\frac{1}{2n+1}\left ( \frac{1}{2} \right )^{n}$. I just thought that this series is the derivation of the original one, but it is not.

Then I know that: $\sum\limits_{n=0}^{\infty}x^{n}=\frac{1}{1-x}$, so in my case $x=\frac{1}{2}$, but I really do not know, how I should get the fraction.

Can anyone please help me?

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    $\begingroup$ $\frac{1}{2n+1}(1/2)^n=\sqrt{2}\cdot \frac{1}{2n+1}(1/\sqrt{2})^{2n+1}$ and the derivative of $\frac{1}{2n+1}x^{2n+1}$ is $x^{2n}$. So, you can start with $\sum_n x^{2n}=\frac{1}{1-x^2}$, integrate, evaluate at $x=1/\sqrt{2}$ and multiply by $\sqrt{2}$. $\endgroup$ – MoonLightSyzygy Jan 20 at 21:47
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Since$$\sum_{n=0}^\infty x^{2n}=\sum_{n=0}^\infty(x^2)^n=\frac1{1-x^2}$$and since$$\left(\sum_{n=0}^\infty\frac{x^{2n+1}}{2n+1}\right)'=\sum_{n=0}^\infty x^{2n}=\frac1{1-x^2}\text{ and }\sum_{n=0}^\infty\frac{0^{2n+1}}{2n+1}=0,$$you have$$\sum_{n=0}^\infty\frac{x^{2n+1}}{2n+1}=\int_0^x\frac{\mathrm dt}{1-t^2}=\operatorname{arctanh}x.$$Therefore$$x\neq0\implies\sum_{n=0}^\infty\frac{x^{2n}}{2n+1}=\frac{\operatorname{arctanh}x}x.$$In particular\begin{align}\sum_{n=0}^\infty\frac1{2n+1}\left(\frac12\right)^n&=\sum_{n=0}^\infty\frac1{2n+1}\left(\frac1{\sqrt2}\right)^{2n}\\&=\frac{\operatorname{arctanh}\left(\frac1{\sqrt2}\right)}{\frac1{\sqrt2}}\\&=\sqrt 2\operatorname{arctanh}\left(\frac1{\sqrt2}\right)\\&=\frac1{\sqrt2}\log\left(\frac{\sqrt2+1}{\sqrt2-1}\right).\end{align}

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  • $\begingroup$ Thank you so much $\endgroup$ – Peter Jan 21 at 9:14
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\begin{align} & \sum_{n=0}^\infty \frac{(1/2)^n}{2n+1} = \sum_{n=0}^\infty \frac{(1/\sqrt2\,)^{2n}}{2n+1} \\[8pt] = {} & \sqrt 2 \sum_{n=0}^\infty \frac{(1/\sqrt2\,)^{2n+1}}{2n+1} \\[10pt] & \frac d {dx} \sum_{n=0}^\infty \frac{x^{2n+1}}{2n+1} = \sum_{n=0}^\infty x^{2n} = \frac 1 {1-x^2}. \end{align} So you need an antiderivative of $$ \frac 1 {1-x^2} = \frac 1 {(1-x)(1+x)} = \frac A {1-x} + \frac B {1-x}. $$ You need to complete the partial-fractions problem. To find the value of the constant of integration, use the fact that when $x=0$ in the sum above, then the sum is $0.$

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