Testing convergence

Question

Test the convergence of the series through the ratio test and the integral test:

∑ (to ∞) n⋅e^(-2n)

n=1

Answer 1

ratio test is to see if limit of a_n+1/a_n is < 1 as n goes to infinity:

Limit as n goes to infinity of ((n+1)/n)(e²ⁿ/e²⁽ⁿ⁺¹⁾) = 1/e² (converges)

Integral test is to test convergence of the series by comparing it with the integral from 1 to infinity of xe^-2x

You have to integrate by parts to get -xe^-2x/2 - e^-2x/4 from 1 to a as a approaches infinity

The limit terms approach 0 and the 1 terms lead to e^-2/2 + e^-2/4 = 3/(4e²)