Question

How do you use the integral test to determine the convergence or divergence of `1+1/(2sqrt2)+1/(3sqrt3)+1/(4sqrt4)+1/(5sqrt5)+...`?

Answer 1

The series:

`sum_(n=1)^oo 1/(nsqrt(n))`

is convergent.

Explanation:

We have the series:

`sum_(n=1)^oo 1/(nsqrt(n))`

we can apply the integral test using as test function:

`f(x) = 1/(xsqrt(x)) = x^(-3/2)`

since the function is positive and decreasing in the interval `(1.+oo)` and we have:

`lim_(x->oo) x^(-3/2) = 0`

Calculating the improper integral:

`int_1^oo x^(-3/2)dx = [-2x^(-1/2)]_1^oo`

`int_1^oo x^(-3/2)dx = 2-lim_(x->oo) -2/sqrt(x) = 2`

As the integral is convergent then the series is proven to be convergent.