How do you use the limit comparison test on the series sum_(n=1)^oo(n+1)/(n*sqrt(n))∞∑n=1n+1n⋅√n ?

1 Answer
Andrea S.
May 18, 2018

The series:

sum_(n=1)^oo (n+1)/(nsqrtn)∞∑n=1n+1n√n

is divergent.

Explanation:

Note that:

(n+1)/(nsqrtn) = n/(nsqrtn) +1/(nsqrtn) = 1/sqrtn+1/(nsqrtn)n+1n√n=nn√n+1n√n=1√n+1n√n

as for every n in NN:

1/(nsqrtn) > 0

it follows that:

(n+1)/(nsqrtn) > 1/sqrtn

and as the series:

sum_(n=1)^oo 1/sqrtn = sum_(n=1)^oo n^(-1/2)

is divergent based on the p series test, we can conclude that:

sum_(n=1)^oo (n+1)/(nsqrtn)

is also divergent.

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