Question

# Determine the convergence or divergence of the series. sum_{n=1}^infty(1+frac{1}{n})^n

Series
Determine the convergence or divergence of the series.
$$\sum_{n=1}^\infty(1+\frac{1}{n})^n$$

2020-12-10
The given series is
$$\sum_{n=1}^\infty(1+\frac{1}{n})^n$$
To test the convergence or divergence first apply series divergence test.
if $$\lim_{n\to\infty}a_n$$ does not exist or $$\lim_{n\to\infty}a_n\ne0\Rightarrow\sum_{n=1}^\infty$$ diverges
here apply series divergence test
$$\lim_{n\to\infty}(1+\frac{1}{n})^n=e$$
Since the limit is not equal to zero
$$\sum_{n=1}^\infty(1+\frac{1}{n})^n$$ diverges