Question

What should I put for using the integral test to determine whether the series is convergent or divergent?

DIVERGES and DIVERGENT doesn't work.

~~~~~~~~~~~~~~~~~

Answer 1

The series:

`sum_(n=1)^oo a_n = sum_(n=1)^oo n/(n^2+5)`

is divergent.

Explanation:

Based on the integral test, the convergence of the series:

`sum_(n=1)^oo a_n = sum_(n=1)^oo n/(n^2+5)`

is equivalent to the convergence of the improper integral:

`int_1^oo x/(x^2+5)dx`

as the function `f(x) = x/(x^2+5)` in the interval `[1,+oo)` is positive, monotone decreasing and:

`f(n) = a_n`

Evaluate the indefinite integral:

`int x/(x^2+5) dx = 1/2int (d(x^2+5))/(x^2+5) =1/2ln(x^2+5) + C`

then:

`int_1^oo x/(x^2+5)dx = lim_(x->oo) 1/2ln(x^2+5) - 1/2ln6 = +oo`

So the series is divergent.