1. Convergences of Series

Suppose are real numbers
what is
for
let for then is called the partial sum
that is

Definition.

suppose is a sequence of real numbers
Then the series is said to converge if the sequence of partial sums converges
that is

If the series doesn't converge It is meaningless

Examples

Here converges therefore so is the series.

A series converges if and only if there exist an such that for every , there exist such that

In terms of Cauchy Sequence

A series converges if and only if for every there exist such that

Theorem.

For with

Proof.
For

using Bernoulli's inequality we can show for

□

Theorem.

Suppose is a convergent series of .
Then

Proof.

using 1. Cauchy Criterion for such that

let

□
Note the converse is not true example does not converge we can show that converges to infinite

Theorem.

let be sequence of
such that and then

Proof.

Claim.

To prove Both converge to same value Hope Sir proves this

Interactive Graph