 
Hello Students,
Dr. Gajendra Purohit
and today we are going to learn about the topic
D'alembert's Ratio Test and it is a very important topic,
If this test is understood by you then in my other topics I am going to take Raabe's Test
and
Logarithmic Test you can easily understand
because its use will be there
and students before this I have already uploaded
the topics that are infinite series
and sequence and series in this conversion and diversion
The two tests are comparison test
and the one test is cochin trigal test
so, students, you can
go through my previous uploaded videos
and the basic concepts on the sequence and series
and also my older videos for the students of higher engineering mathematics and BSC students
which I have uploaded
these are here
you can also view these videos in my playlist
and students if you think my channel is
really helping you then subscribe my channel and press the notification bell
and also share it with your friends, also like my channel
so now look here what the D'Alembert's ratio test is
the series of positive terms
is convergent. This series will be convergent
only when
 
If this value is lesser than 1
then it is convergent and if
the value is greater than 1 then it is a divergence and if the value is equal
the test will fail, and if the test fails you will ask sir now what will we do
when the test gets fail
now when I will upload my next videos
where  I am going to tell you about
Raabe's Test and Logarithmic Test this will be applied there
So for this, watch my next videos also
Here, the concept is used
 
Then the series is convergent, and remember this concept
and this concept we are going to apply here
 
 
then this will be divergent, I will tell you how
So, now look test the convergent of series
where x is positive, so students, first of all, we will write
 
 
So the N term will be n sq.
 
 
 
 
 
 
 
Now we are going to check limit,
no not limit we will talk about
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Now, by D'Alembert
Ratio Test
x is less than 1
Now we will write
 
 
 
 
 
 
Now we got to know
that this value is coming x, and if x is smaller than 1
then this will be convergent
 
 
if equal to 1 then
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
so when the value of x is greater or equal than
it will be divergent
 
 
 
 
 
 
 
 
 
 
 
 
 
 
by this way, we will solve the question
but in some questions, we cannot solve directly
 
in my next videos, I will make you understand the concept and how to apply them
So, students, I will take 1-2 questions more on this topic and then we will end this topic
Now, look at the next 2 questions
and then we will end this topic
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Now, students here what we are going to do is
 
we will check the value
 
 
we will divide this
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Now, by
D'Alembert
Ratio Test
 
convergent
 
if x is less than 1
and divergent
if x is greater than 1
if x is equal to 1 then,
in place of x, we will put 1
 
 
so students,
we can also apply the concept we used in previous questions
now, if we want we can also apply a comparison test here
 
 
If you don't have any idea about comparison test
students, you can go to my older videos and have a look
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
by
Comparision test
 
 
 
 
equal to 1 it is convergent
and it is also  convergent at less than 1
I am writing  here now,
because of no space
 
 
 
 
 
By this concept, we will solve this question
so, students, let's have a look at the last question
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
By
D'Alembert
Ratio Test
 
 
 
 
 
 
 
 
 
 
Now we will take the previous concept
of comparison test
 
 
 
 
 
 
 
 
 
and by
comparison test
 
 
 
 
 
 
 
students, I have also told
if you don't want to use this concept
 
 
 
but I have also told you this concept
because in some books this concept is used
 
So, students, my will videos will be helping you
and if they are helping you then subscribe to my channel
press the notification bell and comment,
Share and like my channel. Thank you so much
for watching my videos
and the last screen which is appearing has my older videos
I have already  posted and you can go through them  in my playlist
Thank you so much for watching my videos
and next, I will upload rest of the videos related to this topic
the 3 - 4 videos to be uploaded. Thank you so much for watching my videos.
Thank You
