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

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Answer 1 · 2015-04-20T09:28:00

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!

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

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