# If 0< a_n <b_n and the series a_n diverges then the series b_n diverges. Is this statement true or false?

May 9, 2018

True

#### Explanation:

This is the definition of the comparison test for series. This statement is most definitely true.

Let ${a}_{n} = {1}^{n}$ and ${b}_{n} = {2}^{n}$. We can immediately see that

$0 < {a}_{n} < {b}_{n}$

And since ${a}_{n} = 1 + 1 + 1 + 1 + \ldots$ which is clearly divergent

${b}_{n} = 2 + 4 + 8 + 16 + \ldots .$ is also clearly divergent (it just approaches infinity quicker).

Hopefully this helps!