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Wednesday, 23 October 2013

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.

 Some 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 (see ψ):
    \frac{1}{1} +  \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \cdots = \psi.   

    In the analysis of convergence of a series , the series of positive terms hold an important place . obviously for a series of positive terms the sum of   n  terms   Sn   goes on increasing as more and more terms are added up .  However it does not guarantee that the sum of infinite number will exceed any prescribed finite number . It may happen that the increase in the sum goes on decreasing as more and more terms are added up and ultimately this increase become negligible i.e. to say that sum get closer to a definite number .