Can I turn this into an integral???

[@taterpun6211]

The sum can be represented in terms of the digamma function: S = -1/3Σ1/k - 1/(k+1/3) = -γ/3 - 1/3(-γ + Σ1/k - 1/(k+4/3-1)) = -1/3(ѱ(4/3) + γ) = -1/3(ѱ(1/3) + γ) - 1 By multiplication formula (z=1/3, k=3): ѱ(1/3) + ѱ(2/3) + ѱ(1) = 3(ѱ(1) - ln3) ѱ(1/3) + ѱ(2/3) = -2γ - 3ln(3) By reflection formula (z=1/3): ѱ(2/3) - ѱ(1/3) = πcot(π/3) = π/√3 So we have the linear relation: ѱ(1/3) + ѱ(2/3) = -2γ - 3ln(3) ѱ(2/3) - ѱ(1/3) = π/√3 ѱ(1/3) = -γ - 3ln(3)/2 - π/2√3 Hence S = ln(3)/2 + π/6√3 - 1 Alternatively you can use the Gauss digamma theorem

[@slavinojunepri7648]

Some serious info for my recollection right there! Excellent 👌

[@zeravam]

Excellent cooperation between discreet and continuous calculus. Very good

[@owl3math]

Thanks! appreciate it :)

[@waarschijn]

Nice solution. I would do the following: Define three series, A,B,C defined as the sums of respectively 1/(3n-1), 1/(3n), 1/(3n+1), each from n=1 to n=N. Then A+B+C=H_(3N)-1 and B=H_N/3 where H denotes the harmonic numbers. Now the series A-C is nice and can be easily computed to be 1-π/(3√3) using the cotangent residues trick from complex analysis. Then we can find the series we want, C-B, as ((A+B+C)-3B-(A-C))/2. From the approximation H_N≈ln N+γ we find lim H_(3N)-H_N=ln 3. So in total we get (ln 3-1-(1-π/(3√3)))/2 which agrees with your answer.

[@owl3math]

Nice 👍 thanks

[@slavinojunepri7648]

Brilliant approach

[@owl3math]

thanks!

[@damyankorena]

I've been working on an integral similar to this one but in the sum reverse the order of the terms and substitute a natural number k instead of 3. So far it doesn't seem to have a nice solution but I'll still try to genralise because I have nothing else to do with my life.

[@אסיףכץ-י8ג]

It seems to be related to the Digamma function, so you might want to check that out.

[@djridoo]

It's a great video ! The only thing I would like to see more is (even if it is simple and short) an explanation of why we can switch the sum and integral. It makes the video more satisfying for me, It kid of "legitimates" the result. thank you :)

[@owl3math]

Thanks DJ! And it’s a good point too 👍

[@LouisLeCrack]

That’s the hardest part, the rest is easy af

[@LouisLeCrack]

⁠@@owl3mathdo real maths bro, please man. It pains me to see someone that isn’t rigorous

[@ADDiOUMAARIR]

That's fascinating ❤❤❤

[@owl3math]

Thanks!

[@Arandomguy1yl]

Nice vid, I got this idea very quickly because of the Putnam 2016 B6 problem there someone had to do a type of similiar thing with summation 1/k2^n+1, very similar x^something technique, sometimes looking for shifting the integral outside of the summation to make it manageable is a good idea

[@owl3math]

Thanks! Yes I like this technique :)

[@ronbannon]

Great job!

[@owl3math]

Hey Ron. Thanks! :)

[@markgraham2312]

It equals approximately -0.14839396162690884587003100526504 .

[@noriega411]

This fucked my brain up ngl, really good video

[@owl3math]

😂😂😂 thanks!

[@MikeMagTech]

Nice job!

[@owl3math]

Hey Mike. Thank you!

[@LouisLeCrack]

This is easy af bro

[@BikeArea]

Well done! 👌 (Maybe you would like to work a bit on your intonation as it sounds a little bit monotonous. But that may be just an interpretation by my own ears. ✌️)

[@owl3math]

Thanks. And I think you’re right. I could use a better tone but it’s not my tendency

[@shivx3295]

This question came in ny exam and i did cause i know the way

[@owl3math]

Nice :)

[@froggggggggggggggggggg]

俺やったら出されても解けんな…

[@froggggggggggggggggggg]

和分で解けた！

[@harrydiv321]

wtf is with the dislikes LOL

[@owl3math]

Hey Harry. Like in the comments? This one is doing well with views and thumbs up percentage is very high but the comment section hates it. 😆😂

[@harrydiv321]

@@owl3math i have the "return youtube dislikes" chrome plugin and this video has 356 dislikes for some reason

[@owl3math]

@@harrydiv321 wow! Ok that's really interesting. I can't tell you for sure that's wrong but in youtube studio it is showing a like % of 98.9%. So it comes out to 183 likes to 2 dislikes. What do you make of it?

[@harrydiv321]

@@owl3math ok that's weird, i guess its the extension

[@owl3math]

Ok good. That’s scary 🫣