Question

Why does the integral test not apply to `Sigma (sinn/n)^2` from `[1,oo)`?

Answer 1

Because `f(x) =(sinx/x)^2` is not decreasing for `x in [1,+oo)`

Explanation:

The integral test for convergence of a series requires that the function `f(x)` such that `f(n) = a_n` has to be positive, decreasing and infinitesimal as `x->oo`

If we choose naturally:

`f(x) =(sinx/x)^2`

we can see that #f(x) is positive and infinitesimal, but not decreasing, so the conditions for the integral test are not met.