Convergent series

• The reciprocals of the positive integers produce a divergent series (harmonic series): • : {1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+{1 \over 6}+\cdots \rightarrow \infty. • Alternating the signs of the reciprocals of positive integers produces a convergent series (alternating harmonic series): • :{1 \over 1}-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots = \ln(2) • The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"; see divergence of the sum of the reciprocals of the primes): • : {1 \over 2}+{1 \over 3}+{1 \over 5}+{1 \over 7}+{1 \over 11}+{1 \over 13}+\cdots \rightarrow \infty. • The reciprocals of triangular numbers produce a convergent series: • : {1 \over 1}+{1 \over 3}+{1 \over 6}+{1 \over 10}+{1 \over 15}+{1 \over 21}+\cdots = 2. • The reciprocals of factorials produce a convergent series (see e): • : \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} + \cdots = e. • The reciprocals of square numbers produce a convergent series (the Basel problem): • : {1 \over 1}+{1 \over 4}+{1 \over 9}+{1 \over 16}+{1 \over 25}+{1 \over 36}+\cdots = {\pi^2 \over 6}. • The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"): • : {1 \over 1}+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+{1 \over 32}+\cdots = 2. • The reciprocals of powers of any n>1 produce a convergent series: • : {1 \over 1}+{1 \over n}+{1 \over n^2}+{1 \over n^3}+{1 \over n^4}+{1 \over n^5}+\cdots = {n\over n-1}. • Alternating the signs of reciprocals of powers of 2 also produces a convergent series: • : {1 \over 1}-{1 \over 2}+{1 \over 4}-{1 \over 8}+{1 \over 16}-{1 \over 32}+\cdots = {2\over3}. • Alternating the signs of reciprocals of powers of any n>1 produces a convergent series: • : {1 \over 1}-{1 \over n}+{1 \over n^2}-{1 \over n^3}+{1 \over n^4}-{1 \over n^5}+\cdots = {n\over n+1}. • The reciprocals of Fibonacci numbers produce a convergent series (see ψ): • : \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \cdots = \psi. == Convergence tests ==

There are a number of methods of determining whether a series converges or diverges. Comparison test. The terms of the sequence \left \{ a_n \right \} are compared to those of another sequence \left \{ b_n \right \}. If, for all n, 0 \le \ a_n \le \ b_n, and \sum_{n=1}^\infty b_n converges, then so does \sum_{n=1}^\infty a_n. However, if, for all n, 0 \le \ b_n \le \ a_n, and \sum_{n=1}^\infty b_n diverges, then so does \sum_{n=1}^\infty a_n. Ratio test. Assume that for all n, a_n is not zero. Suppose that there exists r such that :\lim_{n \to \infty} \left|{\frac{a_{n+1}}{a_n}}\right| = r. If r r = \limsup_{n\to\infty}\sqrt[n], :where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value). If r f(n) = a_n be a positive and monotonically decreasing function. If :\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx then the series converges. But if the integral diverges, then the series does so as well. Limit comparison test. If \left \{ a_n \right \}, \left \{ b_n \right \} > 0, and the limit \lim_{n \to \infty} \frac{a_n}{b_n} exists and is not zero, then \sum_{n=1}^\infty a_n converges if and only if \sum_{n=1}^\infty b_n converges. Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form \sum_{n=1}^\infty a_n (-1)^n, if \left \{ a_n \right \} is monotonically decreasing, and has a limit of 0 at infinity, then the series converges. Cauchy condensation test. If \left \{ a_n \right \} is a positive monotone decreasing sequence, then \sum_{n=1}^\infty a_n converges if and only if \sum_{k=1}^\infty 2^k a_{2^{k}} converges. '''Dirichlet's test''' '''Abel's test''' == Conditional and absolute convergence ==

If the series \sum_{n=1}^\infty \left| a_n \right| converges, then the series \sum_{n=1}^\infty a_n is said to be absolutely convergent. Every absolute convergent series (real or complex) is also convergent, but the converse is not true. The Maclaurin series of the exponential function is absolutely convergent for every complex value of the variable. If the series \sum_{n=1}^\infty a_n converges but the series \sum_{n=1}^\infty \left| a_n \right| diverges, then the series \sum_{n=1}^\infty a_n is conditionally convergent. The Maclaurin series of the logarithm function \ln(1+x) is conditionally convergent for (see the Mercator series). 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. Agnew's theorem characterizes rearrangements that preserve convergence for all series. == Uniform convergence ==

Let \left \{ f_1,\ f_2,\ f_3,\dots \right \} be a sequence of functions. The series \sum_{n=1}^\infty f_n is said to converge uniformly to f if the sequence \{s_n\} of partial sums defined by : s_n(x) = \sum_{k=1}^n f_k (x) 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 :\sum_{n=1}^\infty a_n converges if and only if the sequence of partial sums is a Cauchy sequence. This means that for every \varepsilon > 0, there is a positive integer N such that for all n \geq m \geq N we have : \left| \sum_{k=m}^n a_k \right| This is equivalent to \lim_{m \to \infty} \left(\sup_{n>m} \left|\sum_{k=m}^{n} a_k \right| \right) = 0. == See also==