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experimentX

  • 3 years ago

Prove:- \[ \lim_{N\rightarrow\infty}\sum_{k=1}^N\left(\frac{k-1}{N}\right)^N = {1 \over e - 1}\]

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  1. zzr0ck3r
    • 3 years ago
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    I only know how to prove limits of sequences not series:(

  2. zzr0ck3r
    • 3 years ago
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    what a cool series though..

  3. experimentX
    • 3 years ago
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    yep!! this is very interesting!!

  4. zzr0ck3r
    • 3 years ago
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    god we really dont know what we have on our hands with this math stuff.

  5. Herp_Derp
    • 3 years ago
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    I messed up on my first one... \[\lim_{N\rightarrow\infty}\left(\frac{k-1}{N}\right)^N=\lim_{N\rightarrow\infty}\left(1+\frac{k-1-N}{N}\right)^N=e^{k-1-N}.\]

  6. experimentX
    • 3 years ago
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    you missed the sum. the approach is +ve ... BRB in 3 hrs

  7. Herp_Derp
    • 3 years ago
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    And you can use a geometric series from there to get \(\displaystyle \frac{e^{-1}}{1-e^{-1}}\)

  8. Herp_Derp
    • 3 years ago
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    I know, it's not very rigorous. I've only learned as much analysis as I've taught myself, so I'm still not very good with proofs...

  9. experimentX
    • 3 years ago
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    change of limits on the sum is the trick!!

  10. experimentX
    • 3 years ago
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    sorry.. what do we call that ... k ... counter as in loop

  11. Herp_Derp
    • 3 years ago
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    Are you allowed to take the limit through the sum, as I have done?

  12. Herp_Derp
    • 3 years ago
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    For example, to go from:\[\large\lim_{x\rightarrow\infty}\sum_{k=0}^\infty f(x)\overset{?}{=}\sum_{k=0}^\infty \lim_{x\rightarrow\infty}f(x)\]

  13. experimentX
    • 3 years ago
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    this is equivalent to lim n->inf (1 - 1/n)^n + (1 - 2/n)^n + (1 - 3/n)^n + ....

  14. experimentX
    • 3 years ago
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    let's put a new counter m = N - k + 1 <-- this counts from opposite.

  15. Herp_Derp
    • 3 years ago
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    Yeah, that's what I did too. I didn't know it was called a counter, though.

  16. experimentX
    • 3 years ago
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    i don't know what it is called either ... may be some incremental variable.

  17. experimentX
    • 3 years ago
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    in programming ... in loop ... it's called counter... :D

  18. Herp_Derp
    • 3 years ago
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    lol

  19. experimentX
    • 3 years ago
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    gotta go ... bye

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