Infinite series 7 / ( n^(1+(1/n)) ) from n = 8 to infinity. What test do I need to determine if this converges or diverges?

Question

Infinite series 7 / ( n^1+(1/n) ) from n = 8 to infinity. What test do I need to determine if this converges or diverges? Is ratio test involved or is another test needed? Please show work.

Answer 1

The p-series test. For all values of n, the series can be written as a constant times n ^ (-p) where p is always greater than 1. Even though p changes as n changes, because p is always greater than 1, we can still apply it to determine convergence.

Alternatively we can think of this as an application of the comparison test. We know that n ^ (- (1 + x)) converges for x > 0 no matter how small x is because of the p-series. We also know that this series is smaller than the sum of n ^ (- (1+x)) for very small x, because in the limit of large n, your series behaves the same way as n ^ (- (1 + x)), but for small n, you series is smaller than n ^ (- (1 + x)) because your series has a larger negative exponent.

Hope this helps.