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badreferences

  • 2 years ago

Let \(n\in\mathbb N\). For\[e^xf_n(x)=\sum_{k=1}^\infty\frac{k^nx^k}{\left(k-1\right)!}\]show that \(f_n(x)\) is a polynomial of degree \(n+1\) with integer coefficients. Tricky question.

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  1. badreferences
    • 2 years ago
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    @TuringTest @KingGeorge @Zarkon You guys might be interest.

  2. badreferences
    • 2 years ago
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    @bahrom7893 You too, maybe lol.

  3. TuringTest
    • 2 years ago
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    I got a linear thingy when I tried it that makes no sense, I'll write my work in a minute just so somebody can laugh at it

  4. bahrom7893
    • 2 years ago
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    no im most likely not interested lol

  5. bahrom7893
    • 2 years ago
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    Anyway... I'm off for tonight guys, interviews in 10 hrs. I need my sleep. Gnite eastern front.

  6. badreferences
    • 2 years ago
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    Cheers, good luck. Don't dead.

  7. bahrom7893
    • 2 years ago
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    thanks :)

  8. perl
    • 2 years ago
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    what is f sub n (x) ?

  9. badreferences
    • 2 years ago
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    You can ignore the sub. It's just a marker to show that the function \(f\) is dependent on \(n\).

  10. perl
    • 2 years ago
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    well first lets look at e^x, whats the series of this

  11. badreferences
    • 2 years ago
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    You don't have to walk me through it, lol, I already have the solution. This is just a very difficult challenge.

  12. perl
    • 2 years ago
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    e^x |dw:1348817638823:dw|

  13. perl
    • 2 years ago
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    so we can see immediately that

  14. perl
    • 2 years ago
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    |dw:1348817750800:dw|

  15. perl
    • 2 years ago
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    whats solution

  16. badreferences
    • 2 years ago
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    I'll post it when I can pick it up, Sir, it's not in my possession right now.

  17. perl
    • 2 years ago
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    ohhh

  18. perl
    • 2 years ago
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    darn

  19. perl
    • 2 years ago
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    we know that

  20. perl
    • 2 years ago
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    |dw:1348817984611:dw|

  21. mukushla
    • 2 years ago
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    see if this is right or not http://openstudy.com/study#/updates/50651902e4b08d185211d536

  22. mukushla
    • 2 years ago
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    i proved that\[f_1(x)=x+x^2\]and\[f_{n+1}(x)=x(f_n(x)+f_n^'(x))\]

  23. badreferences
    • 2 years ago
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    Full solution. I can't seem to attach the .ps file so I did a screencap with ghostview,

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spraguer (Moderator)
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