Kaycee Roche

2020-12-09

Determine the convergence or divergence of the series.
$\sum _{n=1}^{\mathrm{\infty }}\left(1+\frac{1}{n}{\right)}^{n}$

delilnaT

The given series is
$\sum _{n=1}^{\mathrm{\infty }}\left(1+\frac{1}{n}{\right)}^{n}$
To test the convergence or divergence first apply series divergence test.
if $\underset{n\to \mathrm{\infty }}{lim}{a}_{n}$ does not exist or $\underset{n\to \mathrm{\infty }}{lim}{a}_{n}\ne 0⇒\sum _{n=1}^{\mathrm{\infty }}$ diverges
here apply series divergence test
$\underset{n\to \mathrm{\infty }}{lim}\left(1+\frac{1}{n}{\right)}^{n}=e$
Since the limit is not equal to zero
$\sum _{n=1}^{\mathrm{\infty }}\left(1+\frac{1}{n}{\right)}^{n}$ diverges

Jeffrey Jordon