Remainder Estimate For The Integral Test

This calculus 2 video tutorial explains how to find the remainder estimate for the integral test. It also explains how to estimate the sum of the infinite series using the partial sum and using the remainder estimation theorem for the integral test and confirming the answer with the calculator. This video also explains how to find the number of terms for the estimate to accurate within 0.005.
Improper Integrals:
• Improper Integrals - C...
Converging & Diverging Sequences:
• Converging and Divergi...
Monotonic & Bounded Sequences:
• Monotonic Sequences an...
Absolute Value Theorem - Sequences:
• Absolute Value Theorem...
Squeeze Theorem - Sequences:
• Squeeze Theorem For Se...
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Geometric Series & Sequences:
• Geometric Series and G...
Introduction to Series - Convergence:
• Convergence and Diverg...
Divergence Test For Series:
• Divergence Test For Se...
Harmonic Series:
• Harmonic Series
Telescoping Series:
• Telescoping Series
__________________________________
Integral Test For Divergence:
• Calculus 2 - Integral ...
Remainder Estimate - Integral Test:
• Remainder Estimate For...
P-Series:
• P-series
Direct Comparison Test:
• Direct Comparison Test...
Limit Comparison Test:
• Limit Comparison Test
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[@TheOrganicChemistryTutor]

Next Video: kzbin.info/www/bejne/rmeVYqWqftyVlZY Final Exams and Video Playlists: www.video-tutor.net/ Full-Length Math & Science Videos: www.patreon.com/mathsciencetutor/collections

[@georgesadler7830]

Professor Organic Chemistry Tutor, thank you for explaining the Remainder Estimate for the Integral Test in Calculus Two. I am somewhat confused on this estimate for the Integral Test; however, doing problems is the best way to increase my understanding of the material. This is an error free video/lecture on KZbin TV with the Organic Chemistry Tutor.

[@muhsinkhalif3621]

thanks man! we appreciate your efforts a lot and God bless you!

[@shayorshayorshayor]

"this might take a hot minute"

[@darkducky2]

"This might take a hot minute" 12:45 lmao you right. thanks!

[@cornetapluspluseleven9296]

what kind of calculator are you using? I entered the expression sum(1 / x^2, x, 1, 1e6) // sum from one to a million in wolfram alpha and it took about three seconds.

[@eduardomoreira7624]

😭😭

[@bobingsonbob8893]

​@@cornetapluspluseleven9296 Or if you know a bit of code and are on a computer, you can write an easy for loop and accumulator and it'll go far faster than any calculator (provided you aren't on a dinosaur of a computer, my time for the exact summation that he was doing was .756 seconds just on a basic ass loop).

[@MultiMathtutor]

In part d) you find that n>200. But in part c) the sum is bounded to within 0.005 (the maximum of 1.649768-1.64522 and 1.64522-1.640677) so in fact only 10 terms are needed. To ensure an error less than 0.005 the remainder estimate says that you need more than 200 terms, but part c) says you need only 10. I think it would be good to explain that n>200 only ensures that the error will be less than 0.005 but it doesn't say how much less. If you were to compute the error in part c) with n=200 it will be much less than 0.005.

[@kyokokirigiri166]

Saving us again god bless❤️

[@hermione._.w]

Why when you raise both sides to the power of negative 1 you don't switch the inequality sign also?

[@JamesBrodski]

Thank you for making this video. It is great!

[@ueda6512]

this guy is amazing!

[@MhamadSoubhiKhalil]

Shouldn't we flip the inequality when we raise both sides by -1? 15:34

[@erikdahlen9140]

no?

[@selfwunder]

@@erikdahlen9140 yeah we should it's basic math

[@spencerkimball7450]

The last inequality is giving me some trouble. 0.005

[@allie7049]

you're right! the sign should have been switched because he took the inverse of both sides of the inequality. so n should be greater than or equal to 200 :)

[@robertoberidojr.435]

I also noticed that one

[@albertlegaspi9614]

Recall 1/(n+1)=200 and n>=199....so to satisfy both it should be n>=200

[@zachb9440]

@@albertlegaspi9614 actually appreciate this explanation - it actually showed the error he made (which was not just forgetting to flip the sign)

[@toddholmes594]

Not my favorite video feels like there could be a better structured explanation. Love your videos btw.

[@DerzTheCheeseStick]

For part D, shouldn't you solve 1/n < 0.005 instead of 0.005

[@kamves8822]

It can go either way, 1/n < .005 , deviding n to the right, then deviding .005 from .005n to the left is the same as .005 < 1/n being powered by a negative 1

[@pheonix6553]

@@kamves8822 If you devidde the left side by n you get 1/n^2 not 1

[@zlatanibrahimovic8329]

for part d the video is wrong, you have (integral from n to infinity) > Rn, then you know that if 0.005 > (the integral from n to infinity) then the error must also be less than 0.005. so work out the integral, assuming it's 1/n, you would work with 0.005>1/n>Rn and this is true for n>1/0.005 = 200, so n > 200.

[@lanefrasier6214]

Would the inequality switch as you take the reciprocal of both sides?

[@zomboy33]

Yes

[@aydengertiser4346]

Cool fact: that infinite series (1/x^2) actually converges to pi squared over 6

[@eduardomoreira7624]

Never knew

[@warwick802]

The inequality doesnt change in part d, if you multiply both sides by n and divide both sides by 0.005 you should get n

[@scitwi9164]

Nice :> Now I can approximate the Zeta function that way :)

[@prateeksharma5539]

15:43 you raised the negative power without changing inequality ? Are you kidding???

[@kaddanki096]

Goated Video

[@baharsalman3439]

Allah razı olsun, God bless you

[@lucasargandona4658]

Why does the inequality sign change around 15:00 ?

[@kasungimhana3640]

Very understandable

[@jordanmain7453]

thank you so much!

[@gaspell]

thank you

[@candacehuynh5942]

thank you :]

[@cringy7-year-old5]

actual answer is exactly pi^2/6

[@robertoberidojr.435]

Isn't it confusing that n should be greater than 200 cause 1/n should be greater than 1/200?

[@druwprods6587]

i love you

[@guymorag8380]

I don't get it

[@katieyoner5522]

you're a fucking gem man

[@PoonsYungv]

Why are yall quoting “hot minute” what’s so crazy about that?

[@김지혜-t8o8z]

목소리 존나 좋다