# Thread: Convergence proof...

1. ## Convergence proof...

Check the convergence of:

$\displaystyle \sum^{\infty}_{n=1} \int^{\frac{sin(n)}{n}}_0 \frac{sin(x)}{x}dx$

Thank you!

2. Originally Posted by Also sprach Zarathustra
Check the convergence of:

$\displaystyle \sum^{\infty}_{n=1} \int^{\frac{sin(n)}{n}}_0 \frac{sin(x)}{x}dx$

Thank you!
As $n$ becomes large the intervals of integration contract about 0, where the integrand is close to 1. So we need only consider the convergence of:

$\displaystyle \sum^{\infty}_{n=1} \int^{\frac{sin(n)}{n}}_0 dx$

When $\sin(n)<0$ the corresponding term in the sum is $|\sin(n)|/n$ and when $\sin(n)>0$ the corresponding term is $\sin(n)/n$

So we need only consider the convergence of:

$\displaystyle \sum^{\infty}_{n=1}\frac{|sin(n)|}{n}$

which I'm pretty sure diverges, but will leave the proof of that to the reader

CB