Find the sum of a series

Question

I'm trying to find the sum of the following series:

$$\sum^\infty_{n=1}\frac {(x-3)^{2n}}{2n}$$

I tried to "convert" it to a simple geometrical series, but with no luck. Has someone any idea?

Thanks for inspiration! My solution: $$\sum^\infty_{n=1}\frac {(x-3)^{2n}}{2n} = \sum^\infty_{n=1}{\frac{z^{2n}}{2n}}=\sum^\infty_{n=1}{\int_0^z{z^{2n-1}}dz}=\int^z_0{\sum^\infty_{n=1}{z^{2n-1}}}=\int^z_0{\frac{z}{1-z^2}}=-\frac12 log(-x^2+6x-8)$$

I was wondering if there are any conditions?

$x\neq4$ and $x\neq2$ and $(x-3)^2\leq1$

@dimitrij Do you know the sum of the series $\sum\limits_{n\geqslant1}\frac{z^n}n$, when it exists? — Did, Apr 26 '15 at 17:31
Hint Remember the mclaurin series of the logarithm. What happens then to your series when $(x-3)^2 < 1$ — Alonso Delfín, Apr 26 '15 at 17:37
Note that the $n$-th term is $\int_0^z t^{2n-1},dt$. So our series is obtained by integrating the geometric series $\sum t^{2n-1}$ term by term. — André Nicolas, Apr 26 '15 at 17:47
An $n$ in the denominator can signal we should integrate something, just like an $n$ in the numerator can indicate we should differentiate something. — André Nicolas, Apr 26 '15 at 18:34

score 4 · Answer 1 · edited Apr 26 '15 at 17:46

4

Hint: $$\log(1-x)=-\sum_{n=1}^{\infty }\frac{x^n}{n}$$

Let $x=x^2$, then: $$\log(1-x^2)=-\sum_{n=1}^{\infty }\frac{x^{2n}}{n}$$

edited Apr 26 '15 at 17:46

Did

279,727

answered Apr 26 '15 at 17:37

E.H.E

23,280

score 1 · Accepted Answer · answered Apr 26 '15 at 17:38

Since for any $|z|<1$ we have: $$ -\log(1-z)=\sum_{n\geq 1}\frac{z^n}{n}, \tag{1} $$ $$ -\log(1+z)=\sum_{n\geq 1}\frac{(-1)^n\, z^n}{n}, \tag{2} $$ (you can check them both by integrating termwise the usual geometric series for $\frac{1}{1-z}$ or $\frac{1}{1+z}$) by just summing $(1)$ and $(2)$ one gets: $$ -\frac{1}{2}\log(1-z^2)=\sum_{n\geq 1}\frac{z^{2n}}{2n} \tag{3}$$ for any $|z|<1$. Now you just need to replace $z$ with $x-3$.

Find the sum of a series

2 Answers2