Sunday, 15 September 2013

About Convergent Series



In mathematics, a series is the sum of the terms of a sequence of numbers.
Given a sequence \left \{ a_1,\ a_2,\ a_3,\dots \right \}, the nth partial sum S_n is the sum of the first n terms of the sequence, that is,
S_n = \sum_{k=1}^n a_k.
A series is convergent if the sequence of its partial sums \left \{ S_1,\ S_2,\ S_3,\dots \right \} converges; in other words, it approaches a given number. In more formal language, a series converges if there exists a limit \ell such that for any arbitrarily small positive number \varepsilon > 0, there is a large integer N such that for all n \ge \ N,
\left | S_n - \ell \right \vert \le \ \varepsilon.
A series that is not convergent is said to be divergent.

Examples of convergent and divergent 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:
    {1\over 1} -{1\over 2} + {1\over 3} - {1\over 4} + {1\over 5} \cdots = \ln(2)
  • Alternating the signs of the reciprocals of positive odd integers produces a convergent series (the Leibniz formula for pi):
    {1 \over 1}-{1 \over 3}+{1 \over 5}-{1 \over 7}+{1 \over 9}-{1 \over 11}+\cdots = {\pi \over 4}.
  • The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"):
    {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.
  • 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}.
  • The reciprocals of Fibonacci numbers produce a convergent series:
    \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 \ a_n \le \ b_n, and \sum_{n=1}^\infty a_n diverges, then so does \sum_{n=1}^\infty b_n.
Ratio test. Assume that for all n, a_n > 0. Suppose that there exists r such that
\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = r.
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 non-negative. Define r as follows:
r = \limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|},
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
\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty,
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 monotone 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.

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