What is a comparison test?

Overview

Comparison test, which is also known as limit comparison test. we use this method to test the convergence of an infinite series.

Convergent sequence– A sequence Sn is said to be convergent when it tends to a finite limit. That means the limit of a sequence Sn will be always finite in case of convergent sequence.

Divergent sequence– when a sequence tends to ±∞ then it is called divergent sequence.

Oscillatory sequence-  when a sequence neither converges nor diverges then it is called oscillatory sequence.

Note– a sequence which neither converges nor diverges , is called oscillatory sequence.

A sequence converges to zero Is called null.

Series

Infinite series- If  is a sequence , then is called the infinite series.

It is denoted by  

Examples of infinite series-

Covergent series –  suppose n→∞ , Sn→ a finite limit  ‘s’  , then the seiresSn is said to be convergent .

We can denote it as,

Divergent series–  when Sn tends to infinity then the series is said to be divergent.

Oscillatory series-  whenSn does not tends to a unique limit (finite or infinite) , then it is called Oscillatory series.

Properties of infinite series

1. the convergence and divergence of an infinite series is unchanged addition od deletion of a finite number of term from it.

2. if positive terms of convergent series change their sign , then the series will be convergent.

3. let   converges to s , let k be a non-zero fixed number  then converges to ks.

4. let converges to ‘l’ and  converges to ‘m’.

Comparison test

Statement- Suppose we have two series of positive terms  then, , where k is a finite number , then both series converges or diverges together.

Proof-

we know that by the definition of limits, there exist a positive number epsilon(ε)

Which is very small. Such that

According to definition( comparison test)

for n>m  , that means

ignoring the first m terms of the series,

we get

there will be two cases-

case-1: is convergent , then

where r is finite number

From (1), 

Therefore  ∑u_n  is also convergent.

Case-2: :∑v_n  is divergent, then

From eq. (1)

Then

From (2)

Hence, is also divergent.

Example: Test the convergence of the following series.

Sol. We have 

First we will find un  and vn the 

And

Here, we can see that, the limit is finite and not zero,

Therefore, and converges or diverges together.

Since is of the form   where p = 2>1

So that , we can say that,

is convergent , so that will also be convergent.

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