Does the sum \sum_{n=1}^\infty\frac{\sqrt{n!}}{(3+\sqrt{1})(3+\sqrt{2})...(3+\sqrt{n})} converge or diverge?

Adrienne Rowe

Adrienne Rowe

Answered question

2022-02-25

Does the sum n=1n!(3+1)(3+2)(3+n) converge or diverge?

Answer & Explanation

Lana Barlow

Lana Barlow

Beginner2022-02-26Added 4 answers

Let
an=n!(3+1)(3+2)(3+n)=1(1+31)(1+32)(1+3n)
Then
log(an)=k=1nlog(1+3k)
In the sum, we have log(1+3k)3k as k, so
log(an)k=1n3k6n as n.
Now, it follows that log(an)logn as n, so for n sufficiently large, we have log(an)logn<2. Therefore, for n sufficiently large, an<1n2, and by the comparison test we can conclude that n=1an converges.

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