Convergent series

"Convergent Series" redirects here. For the short story collection, see Convergent Series (short story collection).
In mathematics, a series is the sum of the terms of a sequence of numbers.
Given a sequence , the nth partial sum S_{n} 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 ε > 0, there is a large integer N such that for all ,
A series that is not convergent is said to be divergent.
Contents
Examples of convergent and divergent series
 The reciprocals of the positive integers produce a divergent series (harmonic series):
 Alternating the signs of the reciprocals of positive odd integers produces a convergent series (the Leibniz formula for pi):
 The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"):
 The reciprocals of triangular numbers produce a convergent series:
 The reciprocals of factorials produce a convergent series (see e):
 The reciprocals of square numbers produce a convergent series (the Basel problem):
 The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"):
 Alternating the signs of reciprocals of powers of 2 also produce a convergent series:
 The reciprocals of Fibonacci numbers produce a convergent series (see ψ):
Convergence tests
Main article: Convergence testsThere 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
However, if,
for all n, , and diverges, then so does
Ratio test. Assume that for all n, a_{n} > 0. Suppose that there exists r such that
If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, 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 nonnegative. Define r as follows:
 where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value).
If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, 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 f(n) = a_{n} 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 at infinity, then the series converges.
Cauchy condensation test. If is a monotone decreasing sequence, then converges if and only if converges.
Raabe's test
Conditional and absolute convergence
For any sequence , for all n. Therefore,
This means that if converges, then also converges (but not viceversa).
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.
Uniform convergence
 Main article: uniform convergence.
Let be a sequence of functions. The series is said to converge uniformly to f if the sequence {s_{n}} 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 Mtest.
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 ε > 0, there is a positive integer N such that for all we have
which is equivalent to
See also
 Convergent sequence
 Normal convergence
References
 Rudin, Walter (1976). Principles of Mathematical Analysis. McGrawHill.
 Spivak, Michael (1994). Calculus (3rd ed.). Houston, Texas: Publish or Perish, Inc. ISBN 0914098896.
External links
 Weisstein, Eric (2005). Riemann Series Theorem. Retrieved May 16, 2005.
Categories: Mathematical series
 Convergence (mathematics)
 The reciprocals of the positive integers produce a divergent series (harmonic series):
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