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The series \(\Large \sum\limits_{n=1}^{\infty}\frac{n!}{n^{n}}\) is

A) convergent
B) divergent
C) unbounded
D) none of these

Correct Answer:

A) convergent

Description for Correct answer:
\(\Large \frac{a_{n+1}}{a_{n}}=\frac{\lfloor\underline{n+1}}{(n+1)^{n+1}}=\frac{n^{n}}{\lfloor\underline{n}}=\frac{n^{n}}{(1+n)^{n}}=\frac{1}{ \left(1+\frac{1}{n}\right)^{n}}\)

\( \Large \therefore \lim \begin{vmatrix}\frac{a_{n+1}}{a_{n}}\end{vmatrix}=1\div \) \( \Large \begin{Bmatrix} \lim\limits_{n\rightarrow \infty} \left(1+\frac{1}{n}\right)^{n} \end{Bmatrix} \) \( \Large = \frac{1}{e} < I \)

\(\therefore\) The series is convergent.

Part of solved Real Analysis questions and answers : >> Elementary Mathematics >> Real Analysis

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