Hello Students, I Dr. Gajendra Purohit and today I am going to discuss
convergent or divergent of the infinite series and before this
I have already discussed the topics about the basics of
this
sequence and series and what is the convergent and divergent or what is the bounded and bounded sequence and the other concepts whatever I have used
Scheming in my first video and then in the second video
I'm gonna discuss comparison test or like the P series Test and is
also known as Alpine Test and I have also talked about it  and you can also have a
look at my previously uploaded videos and students there are many older videos for engineering mathematics
and BSC students, IIT Jam, gate and the
students who are preparing for the RPSC or Rajasthan Govt.
1st-grade exam preparation or other MP - State exam for school teacher the
helpful content many upload
so, students, you can see my videos and If you have not subscribed my channel let, go and subscribe first, and press the notification bell
So that my upcoming videos will be notified to you. So, students, today we are going to discuss what Cauchy's Integral Test is
So what Cauchy's integral test is
here we are going to talk about integration, and from integration, we are going to connect series, so this says
positive monotonic decreasing integrable function,
by seeing the question we have to keep in mind that whether it is in decreasing or increasing form, so it must be decreasing
and if we apply this test, then it is useful
value of n then
summation n is equal to 1 to infinity un is convergent if
Improper integral 1 to infinity of FX DX is convergent
Otherwise, it will be divergent, it means whatever is given
if it is convergent and has a finite value, meaning conversion of integration means it has
a finite value, and if the value is not coming finite and coming infinite
then it is divergent if the value of integration is coming finite than it is convergent
and if the value is coming infinite than it is divergent, so, students here we are talking about the limit and it is going from  1 to infinite
and if our sequence is starting from 2 then it will be 2
and if this is starting with 4 then it will be 4 and this we have to look and here
it is from 1 then it will be from 1 to infinite
because it is infinte key sequence and what is that whether it is starting from 1 or 5 the lower limit
the base will be set and the upper limit is infinite only and the value is
not convergent then it is divergent and based on this the questions comes in the exam  now let's talk about
if the student asks how do we know that we have to apply this test or not and I also see in comment students keep on asking
how do we know that we have to apply this test or not, so it will be asked by name
that use the coach integral test for the convergence of the is right  and if in case it is not asked by name so
we can apply any test of our choice and
also which goes with the question and this is mathematics and it comes only with practice
and without practice, it will not be so good, so keep on practicing, and by seeing the question you will
come to know what you have to use clear. So now have a look
I will give you an example, in my previous video I have talked about P series Test
and if you are looking here
so the n term coming here is 1 upon n square so the value of p is coming to the value of p is
2 which is greater than 1 so this series is convergent by the P series Test
but here we don't have to use P series Test
you have to use Cauchy's Integral Test
you cannot do like this but if someone has a competitive exam or competition exam and you have to identify it
then you can use P series Test and it will be easy and helpful but in the exam, if it is asking by name then
by that concept, we will
identify it so students we will
 
Now we will talk about its fx
 
Now we have got the fx value, we will identify the integration value
 
 
 
 
 
 
 
So by Cauchy's Test
Summation also convergent
so, students, it's very easy
 
You can do integration
if the value of integration is finite then it is convergence and if the value is infinite then it is divergent
so, students, I will 2 -3 questions more on this topic
I used the coach integral test to show the series these is convergent or divergent
so students here what we will do is
that what is the here so the n term here it is  2, 3, 4
so, students, it is not starting with 1 it is starting with 2
 
 
 
 
 
if the value is coming finite then it will be convergent
 
 
so students in place of this x
 
 
 
 
 
Here comes the value of A and B
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
is convergent
By Cauchy's Integral Test
Summation u n is also  convergent
So in this way, we will solve the question, the last question I will take on this topic and then we will end it
so students
look at this last question
 
 
so, students,  it is just helping P series test which we have used
in our previous video
the same it is asking for proof here, so students we will talk about
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
so now what's happening here is I will make you understand
I will clear the concept, let us take the value smaller than 1, let's take the value of p = 0
the value is smaller than 1 as soon as we will keep the p-value 0
 
if here the value of p is greater than 1
 
 
I will make you understand what I have done here is
 
 
 
We will simplify it
 
 
 
Convergent
 
divergent
If P is less than equal to 1
So, students, I believe this video is understood by you, and still, if you have any doubts or queries you can comment here
and you can also my videos with your friends as well
so students stay connected with me for more videos
and now in my next video, I am going to discuss the test
D'Alembet Ratio Test, so what is the D'Alembet Ratio Test we will make a video on this. Thank you for watching my video. Thank You
