Question

Use the Integral Test to determine whether the series is convergent or divergent given `sum 1 / n^5` from n=1 to infinity?

Answer 1

It converges.

The Integral Test says that if `a_n>0` is a sequence and `f(x)>0` is monotonic decrescent and `a_n=f(n) forall n`,

`int_1^(+infty)f(x)dx` exists finite `<=> sum_(n=1)^(+infty)a_n` converges.

We know `int_1^(+infty)1/x^5dx=(-1/(4x^4))_1^(+infty)=1/4`, which is finite, so the series converges.

Notice that we don't know the sum of the series, we just know it converges!!