Question

Determine whether the series converges or diverges. (Answer: converges)

sum from n=1 to infinity of (n^2 + 2^n)/(n + 3^n)

Answer 1

\[\sum_{n=1}^{\infty}(n^2 + 2^n)/(n + 3^n)\]

Answer 2

Divide top and bottom by n^2
(1+2^n/n^2)/(1/n+3^n/n^2)
as n->inf, 1 and 1/n drop off:
(2/3)^n
Use ratio test as n-> inf
a(n+1)/a(n)
=(2/3)=2/3 < 1 so the series converges.