The limit of its terms is By the test for divergence, the series

We determine if an infinite series diverges.

### Definition of Infinite Series

**diverges**. If this limit is a finite number, then we say that the infinite series

**converges**.

### Test for Divergence

In this section, we will learn a simple criterion for the divergence of an infinite series.
The main idea is that in order for an infinite series to converge to a finite value, the
terms in the series must approach zero. We now state this fact in its equivalent
**contrapositive** form.

The limit of its terms is By the test for divergence, the series

The limit of its terms is

By the test for divergence, the seriesThe limit of its terms is By the test for divergence, the series

The limit of its terms is By the test for divergence, the series

**does not apply**. The Test for Divergence is

**inconclusive**in this case.

The limit of its terms is By the test for divergence, the series

The limit of its terms is By the test for divergence, the series

The limit of its terms is By the test for divergence, the series

The limit of its terms is By the test for divergence, the series

### Proof of the Test for Divergence

To prove the test for divergence, we will show that if converges, then the limit, , must equal zero. The logic is then that if this limit is not zero, the associated series cannot converge, and it therefore must diverge. We begin by considering the partial sums of the series, . Since the series converges, the limit of the partial sums exists and equals a finite value, which we will call . In mathematical symbols, where To proceed, we make the following two observations: and Putting these facts together, we can conclude that In the above limit, there is nothing special about the index (it was only helpful in dealing with the partial sums, ) and we can change it to to match statement in the theorem. We have therefore proved the theorem, since if then We will wrap up this section by examining the converse of the Test for Divergence. It was mentioned in the remark following the statement of the theorem above, that if the limit of the terms is zero, the associated series could either converge or diverge. We now present two classic series which exemplify this point. The two series are and In both cases, the limit of the terms is zero, but the first series converges while the second series diverges. We will explore both of these series in future sections (the first in the section on Geometric Series and the second in the section on the Integral Test).