Working through a 200 years old Cambridge Exam problem!

This was asked to students in 1801, but it is still a challenging and fun problem to solve to this day!

Source
See page 5, problem 24

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[macieyid]

One can also integrate s · ln(1 - s) to get ½ (s² - 1) · ln(1 - s) - ¼ s² - ½ s, and calculate at s = -1

[jakem]

This was a great review from my calc 2 course. It’s been a few years so naturally I forgot how to solve this, but I’m happy that I can still understand what’s going on.

[Chandichada]

I cannot believe i am still watching this. It is weirdly compelling.

Please keep going. You are doing a good job!

[juampialarcon]

I didn't want to go to Cambridge anyways

[FoxMcCloudV]

The nice thing about this infinite series is that, since it is absolutely convergent, you can rearrange the terms however you wish without changing the sum.

[HelloWorld-dqpn]

Once you are used to partial fraction descomposition, solving this type of problem is an easy task.

[vishalfgm]

i am currently a class 11th student from india and just studied the concept of sequence and series and telescopic sums and series and i solved it and felt proud

[Ultranger]

Nobody:
Presh every few videos: “Let me introduce you to T E L E S C O P I N G S U M S”

[siralanturing]

When you think you are smart, but a 200 year old problem shows you something different.

[thisguyispeculiar]

This can be solved in a shortcut using a bit of quick thinking.
We can expand the given expression as:

1/2 [ (3-1)/(3*1) - (4-2)/(4*2) + (5-3)/(5*3)
= 1/2 [ 1 - 1/3 - 1/2 + 1/4 +1/3 - 1/5 - 1/4 + 1/6 + 1/5

This is a telescopic series where each term after 1/3 has both addition and subtraction to it (indefinitely long series).
Only 1 and -1/2 remain as 1 and 2 can only occur once in the denominator and that happens in first term already.

= 1/2 [1 - 1/2]
= 1/4

And that's the easier way to solve this. No algebra. No formula. But correct answer.

[sp_danger]

Got to be honest, at first I thought the pattern was an alternating double factorial which really hyped me up. nevertheless it was a good problem and a nice solution. Good job.

[krish]

While the very sight of the problem is so daunting, watching the solution unfold in your video is so interesting. Makes me want to try it out myself.😊

[UECAshutoshKumar]

I wrote the general term as
[-S = *summation {1 to infinity}* (-1)^n/n(n+2)]
A beautiful question!keep up the good work ❤️

[paradoxxie]

"And that's the answer"
Very well done 👍 thanks

[hans-rudigerdrzimmermann]

Dear Presh, I was happy to solve as a theoretical physicist with MSc from Cambridge in 1979 this nice question.
I enjoyed at these times the Higher Algebra book from Hall ( Christ College) and Knight (Trinity College) Cambridge.

[Sahil-mysv]

I did the question orally without copy pen because this question is very easy for those who are in school class 11 and preparing for jee

[SeeTv.]

I've watched enough math videos to predict from the beginning that it's gonna be a telescoping sum xD

[cogitoergosum]

Could you tell us what tools you use to create the animations?

[josephszeyu]

Very elegant solution and beautiful question

[sankarrao]

I solved it before watching the video, it's crazy, I learned a lot from presh videos. Thanks