# Infinite Series – Comparison Test For Convergence of Series | Calculus

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

### 100 thoughts on “Infinite Series – Comparison Test For Convergence of Series | Calculus”

• August 16, 2019 at 1:29 pm

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

• August 16, 2019 at 1:31 pm

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

• August 18, 2019 at 5:46 am

I love you sir

• August 19, 2019 at 2:51 pm

thank u so much!!!!

• August 22, 2019 at 4:30 pm

Keep going Sir.

• August 23, 2019 at 2:09 pm

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)

• August 23, 2019 at 3:20 pm

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

• August 24, 2019 at 9:29 am

Awesome sir

• September 2, 2019 at 2:24 pm

Thankyou sir ji

• September 2, 2019 at 2:25 pm

Iit jam ki preparatione ke liye konsi book use kre ?

• September 2, 2019 at 3:58 pm

Thank you very much sir.

• September 2, 2019 at 6:27 pm

Thank u so much Sir….

• September 4, 2019 at 5:23 pm

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

• September 11, 2019 at 2:39 pm

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

• September 12, 2019 at 11:04 am

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

• September 15, 2019 at 12:42 pm

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

• September 16, 2019 at 3:51 am

Thank you sir. π

• September 20, 2019 at 7:29 am

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

• September 22, 2019 at 4:54 am

thanks sirπππ

• September 22, 2019 at 7:15 am

Thank you so much for explaining everything in simple words ππ

• September 26, 2019 at 1:11 pm

Tkanku sir.. Could you solve some more different models

• September 28, 2019 at 3:37 pm

Thanks sirr

• October 2, 2019 at 2:17 pm

summation 1/n divergent Kaise aatha

• October 2, 2019 at 7:35 pm

sir yah bsc 3 rd year me hai

• October 4, 2019 at 11:04 am

Thank you so much sir…

• October 6, 2019 at 11:59 am

(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

• October 6, 2019 at 4:30 pm

Nice explanation

• October 14, 2019 at 2:50 pm

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

• October 14, 2019 at 5:05 pm

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

• October 15, 2019 at 12:56 am

Sir how you got vn=1/n

• October 15, 2019 at 4:17 pm

Thanks A Lot SirβΊοΈβΊοΈ

• October 15, 2019 at 4:48 pm

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

• October 16, 2019 at 4:15 am

Plz sir making the Divergent test

• October 16, 2019 at 3:04 pm

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

• October 19, 2019 at 8:26 am

Sir you great

• October 20, 2019 at 2:27 am

Thankuuu sir excellent explaination

• October 20, 2019 at 8:58 pm

Great sir

• October 25, 2019 at 3:33 pm

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

• October 27, 2019 at 7:27 am

• October 28, 2019 at 2:38 am

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

• November 3, 2019 at 12:06 pm

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

• November 6, 2019 at 2:56 pm

thanks sir your video helped me a lot in Bsc maths

• November 9, 2019 at 4:15 pm

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

• November 11, 2019 at 11:13 am

thanku sir

• November 12, 2019 at 6:09 am

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

• November 14, 2019 at 8:22 am

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?????

• November 15, 2019 at 12:41 am

Sir Vn chose Krna nhi aa rha???

• November 15, 2019 at 2:21 pm

Thank you sir for help

• November 15, 2019 at 6:37 pm

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

• November 19, 2019 at 3:24 pm

Thank u so much sir!! β£οΈ

• November 20, 2019 at 6:38 pm

Thankyou sir

• November 21, 2019 at 3:54 am

Very useful

• November 21, 2019 at 4:40 am

Sir please make vedios on uniform convergence

• November 21, 2019 at 6:10 am

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

• November 23, 2019 at 11:41 am

Thank you very much sir for clearing our concept.
Our 1st semester exam needed this very much

• November 23, 2019 at 4:10 pm

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

• November 24, 2019 at 1:15 pm

Sir how you write it 1|n^3

• November 24, 2019 at 7:18 pm

Thanks you sir

• November 25, 2019 at 6:33 am

Thanku so much sir…u r god for meπ

• November 29, 2019 at 1:45 pm

Bhut hi super sir ji

• November 30, 2019 at 5:21 pm

Thnq so much sir….

• December 2, 2019 at 4:04 pm

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

• December 2, 2019 at 4:06 pm

Thank u sir ur way of teaching is oswm Sir can u provide ur whatsno so that we clear my drought Sir please helpπππππ

• December 2, 2019 at 4:44 pm

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

• December 5, 2019 at 9:51 am

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

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Thanks

Dr.Gajendra Purohit

• December 6, 2019 at 6:34 am

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

• December 8, 2019 at 3:54 pm

Superb sir

• December 10, 2019 at 5:02 am

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

• December 11, 2019 at 5:05 am

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

• December 11, 2019 at 6:01 pm

thank you sir

• December 13, 2019 at 11:53 am

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• December 13, 2019 at 2:11 pm

thanq so much sir

• December 14, 2019 at 12:26 pm

Excellent work sir

Thanx for helping

• December 16, 2019 at 4:09 am

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

• December 17, 2019 at 12:53 pm

Thank u so much

• December 18, 2019 at 9:40 am

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

• December 20, 2019 at 5:01 pm

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

• December 22, 2019 at 7:57 am

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

• December 24, 2019 at 6:47 am

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

• December 25, 2019 at 10:14 am

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

• December 26, 2019 at 2:23 pm

Nys superb teaching sirβ€

• December 26, 2019 at 5:54 pm

Nice video ππππ

• December 27, 2019 at 2:31 am

Nice

• December 27, 2019 at 2:00 pm

Thank you sir, please provide on abstract algebra

• January 2, 2020 at 11:34 am

Thanx sir

• January 2, 2020 at 4:56 pm

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

• January 7, 2020 at 3:19 am

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

• January 16, 2020 at 10:55 am

Gajab sir

• January 16, 2020 at 9:35 pm

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.

• January 17, 2020 at 3:16 pm

Thank sir

• January 18, 2020 at 6:01 am

• January 19, 2020 at 6:27 am

More helpful Sir ππππππ

• January 23, 2020 at 12:50 am

how to join online classes

• January 24, 2020 at 5:12 pm

Sir please provide on abstract algebra

• January 25, 2020 at 3:42 pm

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

• January 28, 2020 at 2:52 am

Thanks for the videos sir .!!ππ

• January 30, 2020 at 12:14 pm

Sir please make more videos on this topic

• February 1, 2020 at 7:20 pm

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

God bless you πβ£οΈπ