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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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@TuringTest @KingGeorge @Zarkon You guys might be interest.
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@bahrom7893 You too, maybe lol.
TuringTest
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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
bahrom7893
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no im most likely not interested lol
bahrom7893
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Anyway... I'm off for tonight guys, interviews in 10 hrs. I need my sleep. Gnite eastern front.
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Cheers, good luck. Don't dead.
bahrom7893
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thanks :)
perl
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what is f sub n (x) ?
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You can ignore the sub. It's just a marker to show that the function \(f\) is dependent on \(n\).
perl
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well first lets look at e^x, whats the series of this
badreferences
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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.
perl
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e^x |dw:1348817638823:dw|
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so we can see immediately that
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|dw:1348817750800:dw|
perl
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whats solution
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I'll post it when I can pick it up, Sir, it's not in my possession right now.
perl
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ohhh
perl
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darn
perl
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we know that
perl
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|dw:1348817984611:dw|
mukushla
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i proved that\[f_1(x)=x+x^2\]and\[f_{n+1}(x)=x(f_n(x)+f_n^'(x))\]
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Full solution. I can't seem to attach the .ps file so I did a screencap with ghostview,