Divergence Test

The divergence test is one of the easiest ways to test whether a series converges or not, by using the limits of it’s sequence.

Definition §

Let $a_n$ be a sequence of real numbers, and ${\sum }_{n=1}^{\infty }{a}_n$ be it’s respective series.

${\lim }_{n\to \infty }{a}_n\ne 0⟹{\sum }_{n=1}^{\infty }{a}_{n}diverges$

• If the limit does not exist, it still satisfies the inequality, and hence the divergence test cannot be used.
• If ${\lim }_{n\to \infty }{a}_n\ne 0$ then ${\sum }_{n=1}^{\infty }{a}_n$ may converge or diverge, other tests are needed.

The principal of this test works by looking at the values that the sequence approaches as $n\to \infty$. If the values does not equal 0, then the sum of the terms (the series) will keeping increasing (or decreasing if the sequence contains negative terms) , and hence will diverge.