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

DIVERGES and DIVERGENT doesn't work.

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1 Answer
Andrea S.
Jul 2, 2017

The series:

sum_(n=1)^oo a_n = sum_(n=1)^oo n/(n^2+5)∞∑n=1an=∞∑n=1nn2+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)∞∑n=1an=∞∑n=1nn2+5

is equivalent to the convergence of the improper integral:

int_1^oo x/(x^2+5)dx∫∞1xx2+5dx

as the function f(x) = x/(x^2+5)f(x)=xx2+5 in the interval [1,+oo)[1,+∞) is positive, monotone decreasing and:

f(n) = a_nf(n)=an

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∫xx2+5dx=12∫d(x2+5)x2+5=12ln(x2+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.

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