Exact Sum of Series

Solution 1:

Note that we can write the sum as $f(1/3)$ where

$$f(x)=x^{-3/2}\sum_{n=1}^{\infty}(-1)^n\frac{\left(\sqrt{x}\right)^{2n+3}}{2n+3}\tag 1$$

Now, denote the series in $(1)$ by $g(x)$. Then, we have

$$g'(x)=\frac12\sqrt{x}\,\,\sum_{n=1}^\infty (-1)^n x^n=- \frac{x^{3/2}}{2(x+1)} \tag 2$$

Integrating both sides of $(2)$ yields

$$g(x)=-\frac13 \sqrt{x}(x-3)-\arctan(\sqrt{x}) \tag 3$$

Now, simply substitute $g(x)$ in $(3)$ into $(1)$ and evaluate at $x=1/3$. Proceeding, we find

$$f(1/3)=\frac83-\frac{\pi\sqrt{3}}{2}$$

Solution 2:

depending on the geometric series $$\frac{1}{1+x}=\sum_{n=0}^{\infty }(-1)^nx^n$$ $$ \frac{1}{1+x}-1=\sum_{n=1}^{\infty }(-1)^nx^n$$ $$ \frac{-x}{1+x}=\sum_{n=1}^{\infty }(-1)^nx^n$$

$x\rightarrow x^2$ $$ \frac{-x^2}{1+x^2}=\sum_{n=1}^{\infty }(-1)^nx^{2n}$$

multiply by $x^2$ $$ \frac{-x^4}{1+x^2}=\sum_{n=1}^{\infty }(-1)^nx^{2n+2}$$ $$ \int_{0}^{x}\frac{-x^4}{1+x^2}dx=\int_{0}^{x}\sum_{n=1}^{\infty }(-1)^nx^{2n+2}dx$$ $$x-x^3/3-\tan^{-1}(x)=\sum_{n=1}^{\infty }\frac{(-1)^nx^{2n+3}}{2n+3}$$ divide by $x^3$ $$1/x^2-1/3-\tan^{-1}(x)/x^3=\sum_{n=1}^{\infty }\frac{(-1)^nx^{2n}}{2n+3}$$

now let $x=\frac{1}{\sqrt{3}}$ $$\sum_{n=1}^{\infty }\frac{(-1)^n}{(2n+3)3^n}=\frac{8}{3}-\frac{\sqrt{3}\pi}{2}$$

Solution 3:

I believe you can get there from $\tan^{-1}(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}x^{2n+1}$.

Substitute $n=m+1$ to get

$$\sum_{m=-1}^{\infty}\frac{(-1)^{m+1}}{2m+3}x^{2m+3}.$$

Then substitute $x=\frac{1}{\sqrt{3}}$.

Should yield $-3\sqrt{3}(\tan^{-1}\frac{1}{\sqrt3}-\frac13)$.

Or something like that.

Solution 4:

\begin{align} &f(x)=\sum_{k=1}^{\infty}x^{2n+2}=x^4\sum_{k=0}^{\infty}x^{2n}=\frac{x^4}{1-x^2}\\ &F(x)=\int f(x) dx=\frac{-x(x^2+3)}3+\frac{\ln {\frac{1+x}{1-x}}}2=\sum_{k=1}^{\infty}\frac{x^{2n+3}}{2n+3}\\ &G(x)=\frac{F(x)}{x^3}=\sum_{k=1}^{\infty}\frac{x^{2n}}{2n+3}=\frac{-(x^2+3)}{3x^2}+\frac{\frac1x\ln {\frac{1+x}{1-x}}}{2x^2}\\ &G\left(\frac{i}{\sqrt3}\right)=\sum_{k=1}^{\infty}\frac{\left(-\frac13\right)^n}{2n+3}=\frac{-(-\frac13+3)}{-1}+\frac{\frac{\sqrt3}{i}\ln {\left(\frac{\sqrt3+i}{2}\right)^2}}{2\cdot\left(-\frac13\right)}\\ &=\frac83-\frac{3\sqrt3}{i} \ln\left(e^{\frac{\pi i}6}\right)=\frac83-\frac{\sqrt3\pi}2 \end{align}