Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
badreferences
Group Title
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.
 one year ago
 one year ago
badreferences Group Title
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.
 one year ago
 one year ago

This Question is Closed

mukushla Group TitleBest ResponseYou've already chosen the best response.6
take 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\[e^xP(x)=\sum_{k=1}^\infty\frac{k^nx^{k}}{(k1)!}\]
 one year ago

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

mukushla Group TitleBest ResponseYou've already chosen the best response.6
lets 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
 one year ago

sauravshakya Group TitleBest ResponseYou've already chosen the best response.0
@mukushla can u show me the derivative of dw:1348828878035:dw
 one year ago

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

sauravshakya Group TitleBest ResponseYou've already chosen the best response.0
OH....... NOW, I got it.... THANX @mukushla
 one year ago

badreferences Group TitleBest ResponseYou've already chosen the best response.0
http://openstudy.com/study#/updates/50650f91e4b08d185211d261 Solution in last post.
 one year ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.