Question

Using the integral test, how do you show whether `sum 1/n^2` diverges or converges from n=1 to infinity?

Answer 1

Since `sum_1^(+oo)a_n<=int_1^(+oo)a_xdx`, if the integral is convergent, so it will be the series.

Since `int_1^(+oo)1/x^2dx` is convergent because it's like:

`int_1^(+oo)1/x^alphadx` with `alpha>1`, it is convergent also the series.

If you want, you can do the integral, but it is useless!

`int_1^(+oo)1/x^2dx=[x^(-2+1)/(-2+1)]_1^(+oo)=[-1/x]_1^(+oo)=0+1=1`.