Question

Hello,

I can't figure out this last number of series, convergence/divergence tests.

E inf, n=1 (5n)^(3n) / (5^n +3)^n

I tried ratio test but it's inconclusive. Polynomial doesn't work either, because of the exponents.

Thanks

Answer 1

just a gut feel -- it converges,
(5n)^(3n) = ((5n)^3)^n = (125n^3)^n
so the fraction is
((125n^3)/(5^n + 3)]^n
Now, exponentials grow much faster than polynomials, so the inside fraction is less than 1/n^2 for n>10

So, it will converge.