# Test for convergence of series:

Series : Expression of that form in which the successive terms are always according to some
definite rule as

Is called series. Here is called n th term of series.
There are two types of series
1) Finite Series
2) Infinite Series
Finite Series: If the no. of term in any series is finite then the series is called finite
series.
Infinite Series: If the no. of term in any series is infinite then the series is called
infinite series.
Eg. itâ€™s denoted by
Now we discuss about convergence of an infinite series of positive terms.
Convergent Series :
An infinite series is said to be convergent if the sequence of its partial sum &lt;Sn&gt; is
convergent = S (finite).

## Test for convergence of series :

1) Dâ€™ Alembertâ€™s Ratio test :
If be a series of positive terms such that
then
(i) if &gt; 1, will be convergent
(ii) if &lt; 1, will be divergent
(iii) if = 1 may either convergent or divergent
Cauchy n th root test :
If be a series of positive terms such that
= (a real number)
then
(i) if &lt;1, will be convergent
(ii) if, &gt;1 will be divergent

(iii) if =1, may either convergent or divergent
Raabeâ€™s tests :
If be a series of positive terms such that

then
(i) if &gt;1, will be convergent
(ii) if &lt;1, will be divergent
(iii) if =1, may either convergent or divergent
Dâ€™e Morgan and Bertrandâ€™s Test :
If be a series of positive terms such that

then
(i) if &gt;1, will be convergent
(ii) if &lt;1, will be divergent
Cauchyâ€™s condensation test :
If the series is a positive terms such that &lt;f(n) is a decreasing sequence and
if a &gt;1 and is a positive integer, then the series and both converge or diverge
together.

### Gauss tests :

If be a series of positive terms and

Where sequence &lt;Yn&gt; is bounded then
(a) if
(i) is convergent if
(ii) is divergent if
(b) if
(i) is convergent if
(ii) is divergent if &lt; 1