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 S_{n}

Examples of Sequences-

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

S_{n} = <2n-1> = {1, 2, 5, 7, 9,……..}

## Types of Sequence

## Convergent Sequence-

Let S_{n} 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.

|S_{n}-l| < ε n>=m and n belongs to N.

Or it is written as Sn->l and read as S_{n} tends to l

where l is called limit of sequence Sn.

Example- S_{n}=<1/n> is converges to limit 0.

## Divergent Sequence-

A sequence S_{n} is said to diverge if lim Sn-> ∞ or Sn-> -.

For example S_{n}= <2n + 1> diverges to +∞

## Oscillatory Sequence-

A sequence S_{n} is said to an oscillatory sequence, if it neither converges nor diverges.

Sequence S_{n}= <(-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 ∑ u_{n} 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 ∑ u_{n} 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 ∑ u_{n} 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 ∑ u_{n} 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 u_{n} 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 ∑ u_{n} 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 ∑ u_{n} be the series of positive terms

Then

where ρ and δ are positive numbers and a_{n} 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.

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