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 2 years ago
Let \(n\in\mathbb N\). For\[e^xf_n(x)=\sum_{k=1}^\infty\frac{k^nx^k}{\left(k1\right)!}\]show that \(f_n(x)\) is a polynomial of degree \(n+1\) with integer coefficients.
Tricky question.
 2 years ago
Let \(n\in\mathbb N\). For\[e^xf_n(x)=\sum_{k=1}^\infty\frac{k^nx^k}{\left(k1\right)!}\]show that \(f_n(x)\) is a polynomial of degree \(n+1\) with integer coefficients. Tricky question.

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mukushla
 2 years ago
Best ResponseYou've already chosen the best response.6take derivative\[e^x=\sum_{k=1}^\infty\frac{x^{k1}}{(k1)!}\]multiply by x\[xe^x=\sum_{k=1}^\infty\frac{x^{k}}{(k1)!}\]and take derivative like this n times and multiply by x again every time finally we will have something like this\[e^xP(x)=\sum_{k=1}^\infty\frac{k^nx^{k}}{(k1)!}\]

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.6actually better to write\[e^xP_n(x)=\sum_{k=1}^\infty\frac{k^nx^{k}}{(k1)!}\]

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.6lets prove by induction that degree of \(P_n(x)\) is n+1 \[P_1(x)=x+x^2\]lets say degree of \(P_n(x)\) is n+1 and prove that degree of \(P_{n+1}(x)\) is n+2\[P_{n+1}(x)=xe^{x}[e^xP_n(x)]'=x(P_n^'(x)+P_n(x))\] which is from degree 1+n+1=n+2

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.0@mukushla can u show me the derivative of dw:1348828878035:dw

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.6\[e^x(1+x)=\sum_{k=1}^\infty\frac{kx^{k1}}{(k1)!}\]

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.0OH....... NOW, I got it.... THANX @mukushla

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.0http://openstudy.com/study#/updates/50650f91e4b08d185211d261 Solution in last post.
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