, a series
is the sum
of the terms of a sequence
Given a sequence , the nth partial sum is the sum of the first n terms of the sequence, that is,
A series is convergent if the sequence of its partial sums converges. In more formal language, a series converges if there exists a limit such that for any arbitrarily small positive number , there is a large integer such that for all ,
A sequence that is not convergent is said to be divergent.
Examples of convergent and divergent series
- The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"):
- The reciprocals of positive integers produce a divergent series:
- Alternating the signs of the reciprocals of positive integers produces a convergent series:
- The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"):
- The reciprocals of square numbers produce a convergent series (the Basel problem):
- Alternating the signs of the reciprocals of positive odd numbers produces a convergent series:
There are a number of methods of determining whether a series converges or diverges.
Comparison test. The terms of the sequence are compared to those of another sequence . If,
for all n, , and converges, then so does .
for all n, , and diverges, then so does .
Ratio test. Assume that for all n, . Suppose that there exists such that
If r < 1, then the series converges. If then the series diverges. If the ratio test is inconclusive, and the series may converge or diverge.
Root test or nth root test. Suppose that the terms of the sequence in question are non-negative, and that there exists r such that
If r < 1, then the series converges. If then the series diverges. If the root test is inconclusive, and the series may converge or diverge.
The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series.
Integral test. The series can be compared to an integral to establish convergence or divergence. Let be a positive and monotone decreasing function. If
then the series converges. But if the integral diverges, then the series does so as well.
Limit comparison test. If , and the limit exists and is not zero, then converges if and only if converges.
Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form , if is monotone decreasing, and has a limit of 0, then the series converges.
Cauchy condensation test. If is a monotone decreasing sequence, then
converges if and only if converges.
Conditional and absolute convergence
For any sequence , for all n. Therefore,
This means that if converges, then also converges (but not vice-versa).
If the series converges, then the series is absolutely convergent. An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long. The power series of the exponential function is absolutely convergent everywhere.
If the series converges but the series diverges, then the series is conditionally convergent. The path formed by connecting the partial sums of a conditionally convergent series is infinitely long. The power series of the logarithm is conditionally convergent.
The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
- Main article: uniform convergence.
Let be a sequence of functions.
The series is said to converge uniformly to f
if the sequence of partial sums defined by
converges uniformly to f.
There is an analogue of the comparison test for infinite series of functions called the Weierstrass M-test.
Cauchy convergence criterion
The Cauchy convergence criterion states that a series
converges if and only if
the sequence of partial sums
is a Cauchy sequence
This means that for every
there is a positive integer
such that for all
which is equivalent to
- Rudin, Walter (1976). Principles of Mathematical Analysis. McGrawHill.
- Spivak, Michael (1994). Calculus (3rd ed.). Houston, Texas: Publish or Perish, Inc. ISBN 0-914098-89-6.