this post was submitted on 16 May 2024
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  • Show that the infinite multiplication (1+1/1)(1+1/2)(1+1/3)... does not converge.
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[–] siriusmart@lemmy.world 1 points 1 year ago* (last edited 1 year ago)

Hint:

spoilere

Solution:

spoilerzkfcfbzr solved it

i put everything into ln because i was scared of multiplication

https://gmtex.siri.sh/fs/1/School/Extra/Maths/Qotd%20solutions/2024-05-16_telescoping-multiplication.html

[–] zkfcfbzr@lemmy.world 1 points 1 year ago* (last edited 1 year ago)

solutionThe terms can be rewritten as:

(2/1) * (3/2) * (4/3) * ... * ((n+1)/n) * ...

Each numerator will cancel with the next denominator. In total everything cancels, so the answer is the empty product, 1.

...Wait...

Uhm, ignore that. Rather, consider the products we get when multiplying. We get: 2/1. 6/2. 24/6. Etc. That is, we have:

Π (n = 1 to k) (n+1)/n = (k+1)! / k! = (k+1)k!/k! = k+1

k+1 clearly goes to infinity as k → ∞, so our product diverges to infinity.