## Using the definition

Using the definition of the sum of a series, solve the following exercises.

• #### Variant 1

Show that the harmonic series $$\displaystyle \sum_{n=1}^{\infty}\frac1n$$ diverges.

• #### Variant 2

Show that the series $$\displaystyle \sum_{n=1}^{\infty} \frac1{\sqrt n}$$ diverges.

• #### Variant 3

Investigate the convergence or divergence of the series $$\displaystyle \sum_{n=1}^{\infty} \ln\left(1+\frac1n\right)$$.

• #### Variant 4

Let $$\displaystyle \lim_{n\to \infty} a_n = a \in \mathbb R$$. Determine $$\displaystyle \sum_{n=1}^{\infty} (a_{n+1}-a_n)$$.