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You Might Be Making This Mistake

BriTheMathGuy

184,377 views

Often times we can solve really amazing problems by switching the integral symbol and the discrete sum symbol - but are we even allowed to do this?

Let's find out when we can switch the integral and sum! (switch integral and series)

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

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20 Comments

@BriTheMathGuy Creator 3 years ago (edited) Pinned

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@nablahnjr.6728 5 years ago

just switch over to physics and it becomes always true

@shehnazsalahuddin6053 5 years ago

Summation: Integral, you can interchange with me anytime, anywhere!
Infinity: I am about to ruin this man's whole career

@tomkerruish2982 5 years ago

Of course they're equal, that's how we do Borel summation!😁

@BriTheMathGuy Creator 5 years ago

What other math facts do you take for granted?

@anshumanagrawal346 5 years ago

Lol, I was thinking about this for a while, but didn't really know much so couldn't resolve it. Glad to see you talking about it and explaining it so well!

@goodplacetostop2973 5 years ago

Thanks for this video, keep up the great content.

@manucitomx 5 years ago

Thank you.
Easy on the eyes, the ears and the mind.

@rgpnovo 5 years ago

The limit at 3:00 is actually 2^{-x}-lim n 2^{-n x} = 2^{-x} - DiracDelta(x)/ln 2. Having taken the limit using distributions, the sum sign and the integral sign may be interchanged.

@sherpearce 5 years ago

5:40 also called d'Alembert criterion

@isavenewspapers8890 1 year ago

2:43 I urge you to parenthesize the summand.

@kaizoisevil 5 years ago

See, I always would use "sum of integral" and "integral of sum" interchangeably, but that was only if the sum was convergent.

@Jack_Callcott 4 years ago

Hi Bri This has been a great video for me. I have always been confused about this since I did a course on Lebesgue and Daniell integration many moons ago.

@Xero3333 5 years ago

you used the absolute convergence theorem, which is fine but you also could have us the uniformed convergence of f_n. If so the limit of n to inf sum f_n does not depend on x,, thus interchange of limits is allowed and since a integral is nothing else than a limit you are allowed to do this. Also uniform convergence implies absolute convergence. I think this more beatifull approach however great video !

@CAG2 4 years ago (edited)

I do wonder, for that integral you computed, what happens if you take the limit as the lower bound approaches 0 from the positive direction?

@edwarddnewgate5196 5 years ago

The big question is how tall are you ?

@manishrajbaraik9784 5 years ago

wish somebody taught me like this during my college

@Jack-e7i8s 1 month ago (edited)

I love how math is so simple like this. To test whether a function is differentiable on a domain, you prove that the derivative exists for values on that domain. To test whether you can exchange the integral and sum, you prove that the values will always converge.