# Thread: summation of a series

1. ## summation of a series

I'm trying to compute $\sum_{i=0}^{\infty}\frac{1}{(n+1)\pi}$. I know the answer is $\infty$, but not sure how to prove this. can anyone help? thank you.

2. Originally Posted by dori1123
I'm trying to compute $\sum_{i=0}^{\infty}\frac{1}{(n+1)\pi}$. I know the answer is $\infty$, but not sure how to prove this. can anyone help? thank you.

$\sum_{i=0}^{\infty}\frac{1}{(n+1)\pi}=\frac{1}{\pi }\sum_{i=1}^{\infty}\frac{1}{n}$ , and I hope you already know the harmonic series diverges so...

Tonio

3. Personally, I find that Cauchy's condensation test is really nice here. Since $\left\{\frac{1}{n}\right\}_{n=1}^{\infty}$ is positive and non-increasing it is true that the series $\sum_{n=1}^{\infty}\frac{1}{n}$ shares convergence/divergence with $\sum_{n=1}^{\infty}2^n\frac{1}{2^n}$

4. Originally Posted by RockHard
Also to further tonio's example because I need to learn this as well, you can the limit of his solution which can be easily done by taking the derivative

$\sum_{i=1}^{\infty}\frac{1}{n}
$

but first we know by one theorem you can write the sequence, lets called it $a_n$, as a $f(x)$
$\lim_{x\to\infty}\frac{1}{x}
$

take the derivative of this function which is simply

$\ln(x)$

then we have $\lim_{x\to\infty}\ln(x)
$

Which we should know but remembering the graph of this function or plotting out some terms on a graph as X increases for values greater than 0 reaches no finite term, aka positive infinity

Apart from the fact that what you wrote doesn't seem to be true in the general case (though a simmilar asymptotic behaviour here happens, if I understood correctly what you probably meant to say), you have some serious mistakes here: the derivative of $\frac{1}{x}$ is $-\frac{1}{x^2}.\,\,\ln x$ is its antiderivative or primitive function.

Perhaps you tried to come up with a rather sui generis "proof" in the spirit of the integral test, but nevertheless it isn't quite so, and some conditions must be fulfilled in the general case.

Tonio