Question

Does the series converge or diverge?

Sum (1)/(n+sqrt(n)*ln(n)), n-> 1 to infinity.

I guess its answer is found using limit comparison? I'm just not really sure how to apply it.

Answer 1

\[\sum _n^{\infty } \frac{1}{n+\sqrt{n} \text{Log}[n]} \]
Mathematica reports:

1. Sum does not converge
2. The ratio test is inconclusive
3.The root test is inconclusive

Browse over to WolframAlpha.com and paste in the following:
Sum [1/(n + Sqrt[n]*Log[n]), {n, Infinity}]

Answer 2

compare your series to the p series 1/n , where p is 1 and therefore divergent. then we can use limit comparison test lim n-->infinity (the original series)/(1/n) , with this we get that the limit is 1 and thus we know that since the series 1/n diverges the original series will also diverge

Answer 3

Thank you, that helps.