Decide whether the sum is convergent or divergent <munderover> &#x2211;<!-- ∑ --> <mrow

groupweird40

groupweird40

Answered question

2022-05-23

Decide whether the sum is convergent or divergent
n = 0 1 n ! ( n e ) n

Answer & Explanation

floygdarvn

floygdarvn

Beginner2022-05-24Added 12 answers

Suppose that n 1. Observe that
log n ! = log n + k = 1 n 1 log k = log n + k = 1 n 1 k k + 1 log k d t log n + k = 1 n 1 k k + 1 log t d t = log n + 1 n log t d t = log n + n log n n + 1.
Taking the exponential of both sides of the inequality gives
n ! n ( n e ) n e 1 e 1 n 1 n ! ( n e ) n .
Consequently,
1 e n = 1 1 n n = 1 1 n ! ( n e ) n ,
but the series on the left-hand side is the divergent harmonic series, i.e., the series in question must diverge too.

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