### Convergent Series

**Convergent Series :-**"If {Sn} tends to a finite and definite system S, then S is defined to be the sum to infinity of the series and the series is said to be Convergent to the sum S" .

"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

*n*th partial sum is the sum of the first

*n*terms of the sequence, that is,

**Some 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 integers produces a convergent 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 produces a convergent series:
- The reciprocals of Fibonacci numbers produce a convergent series (see ψ):
- 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 .

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