How to determine whether the infinite series converges?

How would you figure out if this series converges?

#sum_(n=0) ^oo 10^n/(n+5)^(2n)#

1 Answer
Apr 16, 2018

It converges.

Explanation:

Consider the series

#sum_(n = 0)^oo 10^n/(n + 5)^n#

By the ratio test, we see

#L = lim_(n->oo) (10^(n + 1)/(n +1 + 5)^(n + 1))/(10^n/(n + 5)^n)#

#L = lim_(n-> oo) (10(n + 5)^n)/(n + 6)^(n + 1)#

#L = lim_(n->oo) (10(n + 5)^n)/((n + 6)(n + 6)^n)#

#L = lim_(n->oo) 10/(n + 6)((n + 5)/(n + 6))^n#

The first part of the limit clearly converges to #0#, thus the entire limit is #0#.

This means that the series #10^n/(n +5)^n# converges. By the comparison test, #10^n/(n + 5)^(2n)# must also converge because the latter series approaches #0# even quicker due to the #2n# in the exponent.

Hopefully this helps!