# Test for Convergence of Series

To understand test for  convergence of a series you have to understand sequence and series and related concepts.

## Sequence

A sequence S is a function whose domain is natural numbers and range is real numbers.
i.e S: N->R. It is common practice to write a sequence S as Sn

Examples of Sequences-

Sn = <2n+1> = {3, 5, 7, 9, 11,……..}
Sn = <2n-1> = {1, 2, 5, 7, 9,……..}

## Convergent Sequence-

Let Sn be a sequence, it is said to converge to a real number, if there exists a positive number \epsilon>0 and an integer m such that.
|Sn-l| < ε   n>=m  and n belongs to N.
Or it is written as Sn->l and read as Sn tends to l
where l is called limit of sequence Sn.

Example- Sn=<1/n> is converges to limit 0.

## Divergent Sequence-

A sequence Sn is said to diverge if lim Sn-> ∞ or Sn-> -.
For example Sn= <2n + 1> diverges to +∞

## Oscillatory Sequence-

A sequence Sn is said to an oscillatory sequence, if it neither converges nor diverges.
Sequence Sn= <(-1)n> neither converges nor diverges.
This sequence oscillates from -1 to +1.

## Series-

A series is basically sum of elements of a sequence or more operations may involve with summation in series.

For example-
3+5+7+9+11+……………………… only summation is involved.
1+(1/2)+(1/3)+(1/4)+(1/5)+……………….. summation and division are involved.
1-(1/2)+(1/3)-(1/4)+(1/5)-(1/6)……………….. summation, subtraction and division are involved.

Test of convergence of a series.

To test weather a series converges to a limit or diverges is called test for convergence of series.
To test a series weather it converges or or diverges there or many methods are available.
If one test fails to recognize weather it converges or diverges another method you have to try.

## D’ Alembert Ratio Test for Convergence –

Let ∑ un be the series of positive terms
Then

1- The series will be convergent, if l>1
2- The series will be divergent, if l<1
3- Test fails if l=1

## Raabe’s   Test for Convergence –

Let ∑ un be the series of positive terms
Then

1- The series will be convergent, if l>1
2- The series will be divergent, if l<1
3- Test fails if l=1

## Cauchy’s Root Test

Let ∑ un be the series of positive terms

Then

1- The series will be convergent, if l<1
2- The series will be divergent, if l>1
3- Test fails if l=1

## Logarithmic Test for Convergence-

Let ∑ un be the series of positive terms
Then

1- The series will be convergent, if l>1
2- The series will be divergent, if l<1
3- Test fails if l=1

## De Morgan’s and Bertrand’s Test for Convergence

Let \Sum un be the series of positive terms
Then

1- The series will be convergent, if l>1
2- The series will be divergent, if l<1
3- Test fails if l=1

## Second Logarithmic Ratio Test for Convergence

Let ∑ un be the series of positive terms
Then

1- The series will be convergent, if l>1
2- The series will be divergent, if l<1
3- Test fails if l=1

## Gauss’s Test for Convergence

Let ∑ un be the series of positive terms

Then

where ρ and δ are positive numbers and an is bounded sequence.
Series converges if ρ >1 and diverges if ρ <=1

## Conclusion-

In this post, I have explained about  sequence, series and their test for  convergence  which is very important topic in mathematics.  Hope you will understand the concepts.