Thread: Power series, term-by-term integration

Evaluate
∞
∑ 1 / [n(n+1) $\displaystyle 5^n$]
n=1
(hint: start with a geometric series, then use term-by-term integration)
===================================

Attempt:
∞
∑ $\displaystyle x^n$ = 1/(1-x) for |x|<1
n=0

Take the indefinite integral of both sides...
∞
∑ $\displaystyle x^{n+1}$ /(n+1) + C= -ln(1-x)
n=0
Put x=0 => C=0
∞
∑ $\displaystyle x^{n+1}$ /(n+1) = -ln(1-x)
n=0

Divide both sides by x^2
∞
∑ $\displaystyle x^{n-1}$ /(n+1) = - ln(1-x) / $\displaystyle x^2$
n=0

∞
∑ $\displaystyle x^{n-1}$ /(n+1) = - [ln(1-x) / $\displaystyle x^2$] - 1/x
n=1

How to continue?

Thanks for helping!

Originally Posted by kingwinner

Evaluate
∞
∑ 1 / [n(n+1) $\displaystyle 5^n$]
n=1
(hint: start with a geometric series, then use term-by-term integration)
===================================

Attempt:
∞
∑ $\displaystyle x^n$ = 1/(1-x) for |x|<1
n=0

Take the indefinite integral of both sides...
∞
∑ $\displaystyle x^{n+1}$ /(n+1) + C= -ln(1-x)
n=0
Put x=0 => C=0
∞
∑ $\displaystyle x^{n+1}$ /(n+1) = -ln(1-x)
n=0

Divide both sides by x^2
∞
∑ $\displaystyle x^{n-1}$ /(n+1) = - ln(1-x) / $\displaystyle x^2$
n=0

∞
∑ $\displaystyle x^{n-1}$ /(n+1) = - [ln(1-x) / $\displaystyle x^2$] - 1/x
n=1

How to continue?

Thanks for helping!

I'd begin with $\displaystyle \sum^\infty_{n=1}x^{n-1}=\frac{1}{1-x},\,|x|<1\Longrightarrow \sum^\infty_{n=1}\frac{x^n}{n}=-\ln(1-x),\,|x|<1$ $\displaystyle \Longrightarrow \sum^\infty_{n=1}\frac{x^{n+1}}{n(n+1)}=(1-x)\ln(1-x)+x-1,\,|x|<1$ , and now just substitute $\displaystyle x=\frac{1}{5}$ ...

Tonio

Originally Posted by tonio

I'd begin with $\displaystyle \sum^\infty_{n=1}x^{n-1}=\frac{1}{1-x},\,|x|<1\Longrightarrow \sum^\infty_{n=1}\frac{x^n}{n}=-\ln(1-x),\,|x|<1$ $\displaystyle \Longrightarrow \sum^\infty_{n=1}\frac{x^{n+1}}{n(n+1)}=(1-x)\ln(1-x)+x-1,\,|x|<1$

How did you get the -1 at the end? Are you sure it is -1, instead of 0?

, and now just substitute $\displaystyle x=\frac{1}{5}$ ...

But if we just substitute x=1/5, we will get $\displaystyle 1/5^{n+1}$ which is not what we want. We need $\displaystyle 1/5^n$

Thanks!

Originally Posted by kingwinner

How did you get the -1 at the end? Are you sure it is -1, instead of 0?

A mistake, indeed. Erase it.


But if we just substitute x=1/5, we will get $\displaystyle 1/5^{n+1}$ which is not what we want. We need $\displaystyle 1/5^n$


This is not problem at all: $\displaystyle \frac{1}{5^{n+1}}=\frac{1}{5}\cdot\frac{1}{5^n}$ ...

Tonio


Thanks!

.