Radius and interval of convergence of a power series, using ratio test, ex#1

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#blackpenredpen #math #calculus #apcalculus

[larsivarramberg]

Very good video! Been struggling with intervals of convergence in particular for a while, this finally made me able to solve some problems!

[catalinajohnson]

I have an exam in an 1 hour and you literally helped me sooo much!! THANK

[VivekKumar-ngdh]

I am from India 🇮🇳 your style of teaching is awesome. Love from India.

[KhanSahab-hfsx]

This is great..the way you deliver your lecture is impressive 👍👍 thank you

[boxxer]

Thankyou for this! I have been struggling with series.

[brianmusema]

You are such a brilliant young man. Love from Uganda

[fredross]

Great video! Thank you for going slow and explaining every step!

[geosalatast]

Thank you very much! Once again you are doing maths with fun :) greetings from Greece!!

[uzdefrederic]

Thank you for posting your nice videos, this is brilliant especially because there is not too much theory here

[RITESHKUMAR-fqjs]

Very, very nice explanation🙏🙏🙏🙏 and very conceptual problem

[SeeJessicaRun]

Thank you…..Weeks of learning this in class, but blackpenredpen gets it in less than 14 minutes

[manniman]

@blackpenredpen I have a question. I'm doing some self study using the book Fundamentals of Differential Equations 6th edition by Nagle. In section 8.2 (the section for this video) Nagle says that the Ratio Test is the reciprocal of what you just showed. That is:

Limit as n--->(infinity) of I a(n)/a(n+1) I = L

where (n) and and (n+1) are the subscripts. He says that if the limit (L) exists then the radius of convergence is (L). There's no mention of any convergence or divergence of the series if L<1 or L>1, or inconclusive if L=1: it simply says that the radius of convergence is (L).

It is a bit tough to write this here but it's on page 428, section 8.2. I learned the ratio test as you just showed in this video when I did Calculus 2 but is Nagle referencing the same ratio test as you did here or is it a different version used for a different conclusion?

EDIT: I found my answer, it's the same statement put another way when the limit in the usual Ratio test as demonstrated in this video is L < 1 (whenever the series converges).

[christopherspencer]

Not all Heroes wear capes, some just hold microphones for many hours.

[dandossantos]

Thanks, dude! this makes a lot more sense now.

[colorfulcalculus]

How about when x=1? The first term will become 0^0 which is undefined

[danandreiluca]

Do you know Master Halanay. He is strong in the force.

[LuizPoublan]

Can you actually include - 1 in the interval? Since it converges there, but doesn't converge absolutely?

[BhajanandArti]

bhai bahut mast samjhaya puppy dene ka mann kar raha tujhe

[siphilipe]

I do not see how at the endpoint 3 it diverges?

[justins]

where would we be without the racial test