 
 
Hello Students!
Myself Dr. Gajendra Purohit
Today I m going to start Cauchy"s Integral Test
Before this, I have uploaded videos
in which I have given the idea of Convergence and Divergence
I have discussed some tests
Like Ratio Test
Comparison Test
Other tests you can see by clicking on the i-tab. For Engineering Mathematics and B.Sc students, I have uploaded more videos
At the end of the video
you will get notifications of more videos
So students, let's see what is Cauchy Root Test
of convergence
Almost all the tests are the same
so students get confused
which test to use? So students, seeing the pattern of the question, you'll understand which test to be applied
In Cauchy Root Test
It will be convergent if
 
If its value is less than 1
then the series will be convergent
If its value is greater than 1
then the series will be divergent
If its value is equal to 1, test fails
In that case, we'll apply another test
I'll let you know in which question
this test will be applied
While in the exam it will be asked by name
that solve the question using Cauchy Root Test.
But if it's not asked to use Cauchy Root Test
then you have to keep in mind that
series with power n or n square
so that when you'll take 1/n
the power will get canceled. Such questions
are solved with this method
whose convergence we test
that whether it is convergent or divergent
I'll take 2-3 more examples
 
 
you'll understand by the pattern of question
you have to practice various questions to understand which test to be applied
Let's see how we will solve this question
 
So this is our Un
Write the value of Un here
 
 
 
 
 
 
 
whatever value comes, we'll put the limit
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
this value is equal to e
so answer will be e/3
and we know
e/3 < 1
 
 
 
The series is convergent
 
As I told you earlier, if its value is
less than 1
then the series is convergent and if its value is greater than 1
the series is divergent.
 
So now let's see another question.
 
nth term is given here
you need to find out whether the series is convergent or divergent.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
So let's see the last question here
 
You have to check the convergence of this series
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
So like this, we use Cauchy Root Test
Next we will start
Raabe Test. Then we will take
Logarithmic Test and then we will talk about Alternating Series
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