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Filte1943

Novice

Registered: 2025-04-10

Posts: 1

Convergence of sequences containing factorials and powers

I am trying to find out if the following infinite series converges

Sum from n = 1 to infinity of (n factorial divided by n to the power of n).

I have tried some tests like ratio test and nth root test, but I am not sure if I am doing it right.

For example, with ratio test, I take the n+1th term divided by the nth term, but the expression is quite confusing and I do not know how to simplify or evaluate the limit.

I also tried to think of checking the nth root, but the expression nth root of n factorial divided by n to the power of n is not easy to handle. Has anyone encountered this kind of problem and can help me analyze whether this series converges or diverges?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,763

Re: Convergence of sequences containing factorials and powers

hi Filte1943

Welcome to the forum.

I'm not very confident with series limits but I'll have a go.

The ratio test is

Where a(n) is the general term of the series.

If L < 1 then the series converges.

In your case:

This will tend to 1 from below. The ratio test is inconclusive in such a case.

LATER EDIT: Found this: https://math.stackexchange.com/question … o-infinity

Bob

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