Question

(b) What is a convergent series? What is a divergent series? A series is divergent if the nth term converges to zero. A series is convergent if it is not divergent. A convergent series is a series for which lim n → [infinity] an exists. A series is convergent if it is not divergent. A series is convergent if the sequence of partial sums is a convergent sequence. A series is divergent if it is not convergent. A series is convergent if the nth term converges to zero. A series is divergent if it is not convergent. A series is divergent if the sequence of partial sums is a convergent sequence. A series is convergent if it is not divergent.

Answer 1

Answer:

Convergent and divergent are explained with examples.

If limit exist and partial sum converges or individual term approaches zero then series is convergent otherwise divergent and further checked by methods explained below.

Step-by-step explanation:

Given:

Explanatory Question on convergent  and divergent series

To Find :

What are convergent and divergent series?

Solution:

Convergent series:

A series said to be convergent when  the limits of the series converges to the finite possible value for the series.

Consider a series

Sn=a1+a2+a3+a4+........+an

this form a new series as ,

Sn(n=1 to infinity)=$\lim_{n \to \infty} a_n$

=s.

It is important that, partial sum of series, there should be a limit existed and that is finite in nature then only series converges.

The geometric series gives proper idea how limit decides the convergent and divergent nature.

A series in which the individual term ,approaches zero then series is convergent in nature but not for all time.

The series,

1. A convergent series is series for which limit exist.
2. If the sequence of partial sum is convergent sequence.

G(r,c)=$\frac{c}{(1-r)}$  r+c

where r≠1 then series is converges.

Divergent series:

A series is  infinite series with which does not  converges at  any point ,

had a infinity sequence of partial sums  with no finite limit.

A series where individual sum does not approaches zero diverges.

For e.g.

$\lim_{n \to \infty} (-1)^n$

This series diverges.

Or harmonic series which diverges.

Where is limit tends to infinity ,eventually series added up to infinity means no exact answer will be there.

For summing the divergent series  there method as :

1. FFT
2. extrapolation methods.
3. Abelian theorems
4. Regularity,linearity and stability test for the series.

If these test are checked then we will get to know that which series diverges and converges.