This concept can be proved by various ways, here will discuss about two tests: comparison test and integral test.

Integral Test:Let
us prove the harmonic series diverges by comparing its sum with an
improper integral. Consider an arrangement of rectangle, in which each
rectangle is $\frac{1}{n}$ units high and 1 unit wide.

The total area of the rectangles = Sum of the harmonic series

Area of Rectangle = 1 +

$\frac{1}{2}$ +

$\frac{1}{3}$ + .....

Area under the curve (marked in blue) = $\int_1^{\infty}$ y dx

= $\int_1^{\infty}$ $\frac{1}{x}$ dx = $\infty$

=> Total area of the rectangles = $\infty$

=> $\sum_{n=1}^k$

$\frac{1}{n}$ > $\int_1^{k+1}$

$\frac{1}{x}$ dx = ln(k + 1)

Comparison Test:

Compare harmonic series with the another divergent series.

1 +

$\frac{1}{2}$ +

$\frac{1}{3}$ +

$\frac{1}{4}$ +

$\frac{1}{5}$ +

$\frac{1}{6}$ +

$\frac{1}{7}$ +

$\frac{1}{8}$ + ................

> 1 +

$\frac{1}{2}$ +

$\frac{1}{4}$ +

$\frac{1}{4}$ +

$\frac{1}{8}$ +

$\frac{1}{8}$ +

$\frac{1}{8}$ +

$\frac{1}{8}$ + ................

Just about every term with the harmonic series is in excess of or maybe adequate to the actual equivalent term with the second series, and then the amount of the actual harmonic series have to be in excess of the sum of the 2nd series. Nevertheless, the sum of the 2nd series is infinite:

1 +

$\frac{1}{2}$ +

$(\frac{1}{4} + \frac{1}{4})$ $(\frac{1}{8} + \frac{1}{8}+\frac{1}{8}+\frac{1}{8})$ + .............

= 1 +

$\frac{1}{2}$ +

$\frac{1}{2}$ +

$\frac{1}{2}$ +

$\frac{1}{2}$ + ..........

= $\infty$

Which shows that series is diverges.

Comparison test shows that $\sum_{n=1}^{2^k}$

$\frac{1}{n}$ $\geq$ 1 +

$\frac{k}{2}$ $\forall$ k$^+$.