Convergence

Convergence is a property exhibited by limits, sequences and series. A series is convergent if the sequence of its partial sums tends towards some limit, where the partial sums of a sequence may be defined as follows:

Using summation notation, an infinite series can be expressed as the limit of the partial sums, or:

Then, if

,

where S is a real number, the series, , converges to S. If the limit does not exist, or is not finite (±∞), the series diverges.

Series tests

There are a number of tests that can be used to determine whether a series converges or diverges without the need to find the limit of the sequence of partial sums.

Test type	Condition(s) for convergence	Condition(s) for divergence
nth-term test for divergence	-	If
Geometric series:	If 0 < |r| < 1	If |r| ≥ 1
p-series:	If p > 1	If p ≤ 1
Integral test; f must be positive, continuous, and decreasing	If converges	If diverges
Alternating series:	0 < an+1 ≤ an and	If an ≥ an+1
Direct comparison of an; an > 0	If an ≤ bn and converges	If bn ≤ an and diverges
Limit comparison; an > 0	If exists, is positive, and converges	If exists, is positive, and diverges
Ratio test	If	If If , the series diverges by the nth term test

Tips for using the series tests

The following list is a general guide on when to apply each series test.

• Try the nth term test first. If the nth term does not approach zero, the series diverges. If the nth term equals zero, the test is inconclusive, and another test must be used.
• Determine if the series is a geometric series or a p-series. Generally, it is easiest to determine the convergence/divergence of these types of series. If the series is either of these types of series, apply their respective tests.
• If part or all of the nth term of the series contains a (-1)ⁿ term, the series is likely to be an alternating series. If it is, apply the alternating series test.
• If the nth term in the series can be integrated, use the integral test.
• If the series cannot be integrated but can be compared to another type of series, such as a geometric or p-series, use one of the comparison tests.

Note that the above tests (with the exception of the geometric series test) can only be used to determine whether a series converges or diverges and do not provide the sum in the case that the series converges. If a geometric series converges, the sum can be determined as:

Examples

Determine whether the following series converge or diverge:

1. 
2. 
3. 

i. Since , the series diverges by the nth term test.

ii. The series is a p-series. Since p = 3 > 1, the series converges by the p-series test. It is also possible to use the integral test in this case. Let . f(x) is a positive, continuous, and decreasing function for x ≥ 1, so we can apply the integral test:

Since the integral is a real number, this confirms that the series converges by the integral test. However, it is important to note that ½ is not the sum of the series.

iii. Directly comparing the series with the convergent p-series from example ii., we find that

since (n³ + 2) > n³ for all n. Thus, converges by the direct comparison test.

Absolute convergence

A convergent series is said to converge absolutely if the sum of the absolute values of its terms also converges. For example, the alternating series converges by the alternating series test, and is a convergent p-series. Thus, the series converges absolutely.

In cases where the sum of the absolute values of the terms of a convergent series diverges, the series is said to converge conditionally. For example, the series converges according to the alternating series test. However, is a divergent p-series since p = 1. Thus, the series converges conditionally.

Example

Determine whether the following series converges absolutely or conditionally:

The series is an alternating series and it converges by the alternating series test since,

,

and for all values of n.

Comparing directly to the divergent p-series , the series diverges by the direct comparison test since for all values of n. Thus, our original series converges conditionally.

It is worth noting that a conditionally convergent series does not always act in accordance with the basic rules of arithmetic. Consider the alternating series for ln(2):

We can rearrange the above terms as follows,

then simplify the rearranged series as:

Then, multiplying both sides of the rearranged series by 2 yields

such that the right-hand side of the equation is again ln(2), meaning that the above equation can be rewritten as,

,

which is not a true statement. Manipulating the terms in an attempt to obtain 2ln(2) on both sides of the equation did not work in this case due to the fact that the series for ln(2) is conditionally convergent.