Divergence is a property exhibited by limits, sequences, and series. A series is divergent if the sequence of its partial sums does not tend toward some limit; in other words, the limit either does not exist, or is ±∞. The partial sum of a sequence may be defined as follows:
|S2||=||a1 + a2|
|S3||=||a1 + a2 + a3|
|Sn||=||a1 + a3 + a3 + ... + an|
Using summation notation, an infinite series can be expressed as the limit of the partial sums, or:
where S is a real number, the series, , converges to S. Otherwise, if the limit does not exist, or S is ±∞, then the series diverges. For example, the partial sum of a series is shown below:
Since as , , and the series diverges.
In some cases, it is not necessary to compute the limit to determine whether a series diverges; there are tests for series of a certain type or form that simplify the process of determining convergence and divergence.
The table below shows the various tests that can be used (in place of having to compute the limit of a sequence of partial sums) under certain conditions to determine the convergence or divergence of a series.
|Test type||Condition(s) for convergence||Condition(s) for divergence|
|nth-term test for divergence||-||If|
||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|
||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|
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:
Determine whether the following series converge or diverge:
i. Since , the series diverges by the nth term test.
ii. The series is in the form of a p-series. Since p = 1, the series diverges by the p-series test. It is also possible to use the integral test in this case. Substituting x for n:
This confirms what we found with the p-series test: the series diverges.
iii. From example ii. above, we found that diverges. Using the direct comparison test, when n > 7, so also diverges.