Question

How can I show this series converges?

Answer 1

\[\sum_{n=1}^{∞}\frac{ n^3+\ln(n) }{ \sqrt{n^7+n^2} }\]

Answer 2

\[ \frac{ n^3+\ln(n) }{ \sqrt{n^7+n^2} } = \frac{ n^3}{ \sqrt{n^7+n^2} } +\frac{ \ln(n) }{ \sqrt{n^7+n^2} } \]
log(n) < n and n^7 > n^2
the last term would converge and the first term does not.

Answer 3

When I use my computer it said that both term converges.

Answer 4

I have i find the result by hand...

Answer 5

i looked at this for a while and i guess you could take the limit as n approaches infinity of both; using l'hoptial's rule to deduct that the denominator is always bigger than the numberator for the first n^3.5 > n^3 giving limit of 0 [convergent] and then that ln(n) will turn to 1/n making the limit = 0 [convergent]

i can go into details but you don't sound like you need them