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)#
How would you figure out if this series converges?
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
This means that the series
Hopefully this helps!