Hello students, I Dr. Gajendra Purohit Today we are going to discuss sequence and series in that we will talk about what is a comparison test, and with that how we come to know series is convergent or divergent Previously we discussed about the basic concept of convergent and divergent we discussed the basic terminology and about the formation you can click on the information tab and check out the video there are more videos based on engineering mathematics DSC etc. you can check out the videos you can see the playlist too and do subscribe to my channel and press the notification bell Share to your friends and dont forget to like my video So students, look here what is the comparison test If we have any sequence or series here the positive term is very important and k should be finite positive another point is ultimately both the points are almost the same only if we want to calculate the convergence of un we will decide a vn we will take vn depending on un vn is a series on vn we will test, on that ratio if its value is finite then we will talk about what is the behavior of vn if it is convergent then our series will also be convergent if this value is finite and if our vn is divergent then our un is also divergent then you will have a question, that this vn how we will know it is convergent or divergent so, student, there is a P-series test or helping test here if the vn has this type of series here, the p is power and if it is greater than 1 then the series is convergent if the power is less than 1 then it is divergent what does this mean? if you are given any question in an exam there will be a sequence, this is un looking at the un you have to take the n term of vn series then you will check its ratio that the value is finite then with this helping test, we will come to know if vn is convergent then un is also convergent and if it is divergent then un will also be divergent but for that, this value should be finite let us take an example to understand this concept and how to implement it so as you can see the value of un we have as now we will see how we will calculate vn let us take one more example this is given we will see the highest power of n we will calculate its difference then our vn will be 1/n if in case we have so here depending on un you will choose vn so here it is coming now what we will do we will check its ratio put the values as follows we will simplify it we will check its limit taking n common solve as follows we will keep n as infinite value is as follows and it is positive that is it is finite and positive it means we are implementing the test here now we will see this vn is convergent or divergent we need to know the behaviour of vn you can take it as shown now we will see we will check vn here we have power p here it is 1 by the helping test we conclude the vn is divergent and if are vn is divergent then by comparison test un is divergent so students let me explain you one more time first of all you will be given a sequence take the n term of that depending on this we will calculate its vn we will calculate limit and if the value is finite then we will check by helping test that vn is convergent or divergent and here it is coming divergent series is as follows and value of p is 1 it is coming 1 and at 1 it is divergent so our vn will be divergent and if vn is divergent then un is also divergent if this value is finite then we will decide on vn if it is divergent then un is also divergent let us take one or more examples here so students check out one more question here we have to test the convergent so first we will check it’s un solve as follows we will check what value we are getting you will have to check how to take its n term you just have to solve it like this these are odd so odd is 2n+1 or 2n-1 in place of n we will keep 1 now what we will do we have discussed how we will choose vn there 3n in denominator and numerator it is 1 so it will be solve as follows this needs to be kept in mind now what we will do we will calculate limit so the value of un is we will divide it we will take it common we will put the limits then the answer will be it will be 2, which is finitie and positve as it is finite and positve which means now we can see it is convergent or divergent what we will do is we will check by using helping test when p is greater than 1 then the series is convergent so students, this given series we have is convergent so this is the way we will check it let us see more questions on the topic so students look at these two questions you have to check that this series is convergent or divergent so what we will do is in these type of questions, we will first rationalize in order to remove its under root so after rationalizing simplify it as follows we will be left with we will choose vn we have discussed the method you have to check the degree of n we will solve as follows so its value will be now we will check the limit solve as follows we will simplify it where there is n we will keep it infinite now we will check vn is convergent or divergent so un will also be convergent let us take one last question we are given the n term you have to find out the given series is convergent or divergent whose n term is given so now what we will do we have a formula sin series what is it let us see simplify as follows if we talk about this term so here the degree of n is we have to see the lowest degree now we will calculate limit we will divide this solve it as follows simplifying it after putting limit this will be 0 then our value will be which is finite and positive now we will check series vn is convergent so by comparison test un is convergent so, students, I hope you understood this topic and if you like it, please subscribe to my channel and share it with your friends on the last screen, you can click on the videos

Sir, (n+1)x^n/n^2. Isme vn kese find krenge

Sir, 1/n^1+1/n. Denominator me highest degre kya h vn find krne ke liye

I love you sir

thank u so much!!!!

Keep going Sir.

Your concept too are not clear man…as seen many times.

For ex-if Vn is cgt then Un is cgt

But if Un is dgt then Vn is dgt ..

And too other logical error in different questions..

Remember next time……(just a request)

Sir, if lim un/vn comes to be infinite, then

Awesome sir

Thankyou sir ji

Iit jam ki preparatione ke liye konsi book use kre ?

Thank you very much sir.

Thank u so much Sir….

Sir you are the life saver for me sir me to tension me aagaya tha maths pad ke par aapki videos se mujhe sab topics easy lagne lag gaye hai thanks alot sir

sir in last question Un is 1n^2 so Vn will be 1n. you have taken both same, plz help

sir you are not checking whether vn is greater or smaller than un

sir agr kisi case mein Un/Vn infinite and positive ya negative or finite and negative aaya to kya krna h ?

Thank you sir. π

Sir bsc k exam me theory bhi aata h to sir pls theory bhi kara digiye sir

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Thank you so much for explaining everything in simple words ππ

Tkanku sir.. Could you solve some more different models

Thanks sirr

summation 1/n divergent Kaise aatha

sir yah bsc 3 rd year me hai

Thank you so much sir…

(1/X )+ (1/x-1) + (1/X+1) +(1/x-2) + (1/X+2) + (1/x-2)+……. Examine the convergence of this series.please solve this question and send it's full solution to 6375329150 please sir..thanks

Nice explanation

I understood very clearly after watching ur videos plz send me the link of nites

Sir apki padhai me wo Jan hai jo aur koi nahi de sakta.π

Sir how you got vn=1/n

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Sir in last question why you taken Vn=1/n2

Plz sir making the Divergent test

Sir if un = n logn then what will be vn ?

Sir you great

Thankuuu sir excellent explaination

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sir i salute to you .you are great person .sir you have great knowlege and way of method of teachng is very good.sir i belong pakistan but like u so much .sir plz tel me when do we apply any test

Sir please make me your assistance……

Sir, Plz make video on how to make program on convergence and divergence on scinotes

Tomorrow is my maths exam thank-you Sir it's very helpfull

thanks sir your video helped me a lot in Bsc maths

last question's answer will be divergent.sir,am I right?

thanku sir

Thank u so much sir really your way of teaching is world best

Sir kahi aisa to nhi ki Un ke numerator aur denominator me bounded above aur bounded below jo bhi diya ho uski help se Vn ka selection karte hai?????

Sir Vn chose Krna nhi aa rha???

Thank you sir for help

Common lete samay thoda error aya hi sir root ke ander 1/n^4 lena hi

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Thankyou sir

Very useful

Sir please make vedios on uniform convergence

Thank u very much, sir. Sir, aap bahut achha padhate hai

Thank you very much sir for clearing our concept.

Our 1st semester exam needed this very much

Itne adv kon lagata h yar dimg khrb ho gya video dekhte dkehte

Sir how you write it 1|n^3

Thanks you sir

Thanku so much sir…u r god for meπ

Bhut hi super sir ji

Thnq so much sir….

Sir aap copy karke padha rahe ho vishwakarma sir ki same question krwa rahe ho jo unhone krwaye or waise hi krwa rahe ho . Kisi ka content chori mt kro apne khud k question lao or usse solve kra k dikhao unke ques dekh k or ynha paste kr rahe ho saari videos m aapne aise hi kr rkha hai

…….so please be real don't fake and don't copy

Thank u sir ur way of teaching is oswm Sir can u provide ur whatsno so that we clear my drought Sir please helpπππππ

Ty sir for this help,,,,,sir plzz beta gamma function and vecter space pda do

Thanks for your comment, your appreciation always motivate us to do such kind work.

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In question no 3 P is 1 not 2 so it is divergent

Superb sir

Sir (a+nx)^n/n! Btao plz sir

Sir thank you so much you are very good teacher for me in the world plz sir net ki taiyari ki lie vidio bnae

thank you sir

πππππππ

thanq so much sir

Excellent work sir

Thanx for helping

One suggestion: kindly use a white board… constant noise of chalk hitting the board is really annoying .

Thank u so much

How n square canceled I'm not getting ….its cancelled so 1+ 1/n power 4 will come na

Thank you sir I am study on your channel without any coughing

Vn kaise find Karen????? Sir please explain

Sir…Aap aapp Maths Ke Gawd ho sir….love you sir…thank you for helping me out….π₯³π₯³

people have a doubt about limiting process.1/n=0 so sab ko 1/n form mein lane ki koisis karo

Nys superb teaching sirβ€

Nice video ππππ

Nice

Thank you sir, please provide on abstract algebra

Thanx sir

Sir In the semi Last Example may be The Value of p is 1.

Explain in English I cant understand Hindi, I am Telugu one

Gajab sir

thank you sir, great video. I think the last sum could have been solved by just using the property lim x -> 0 sinx/x, assuming x = 1/n.

Thank sir

Sir paraboloid cone padhado

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how to join online classes

Sir please provide on abstract algebra

Sir kese pta kare ki konsi series me kon sa test apply hoga sir plz help me

Thanks for the videos sir .!!ππ

Sir please make more videos on this topic

Thank you very much sir for making such understandable easy maths videos

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Sir. not vision well on the black boad.

Thanks Sir for upload videos