Question

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

Answer 1

Because the condition only applies if `f(x)` and `a_n=f(n)` are non-negative and monotone decreasing, which is not the case here.

Explanation:

In fact, if we choose:

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

we have that the function is continuous and non-negative for `x>0`, but as:

`f'(x) = frac(xcosx-2-sinx) (x^2)`

is positive for some values of `x`, clearly `f(x)` is not monotone decreasing.

graph{(2+sinx)/x [-10, 10, -5, 5]}