Question and Answers Forum

All Questions      Topic List

Permutation and Combination Questions

Previous in All Question      Next in All Question      

Next in Permutation and Combination      

Question Number 23 by user1 last updated on 25/Jan/15

$$\mathrm{Sum}\mathrm{the}\mathrm{series}\underset{n=0}{\sum ^{\mathrm{\infty }}}{\mathbf{P}}_r\left(n\right)\frac{x^n}{n!},\mathrm{where}{\mathbf{P}}_r\left(n\right)$$$$\mathrm{is}\mathrm{a}\mathrm{polynomial}\mathrm{of}\mathrm{degree}r\mathrm{in}n.$$

Answered by user1 last updated on 31/Oct/14

$${\mathrm{P}}_r\left(n\right)=a_0+a_{1}n+a_{2}n(n-1)+....+a_{r}n(n-1)...(n-r+1),$$$$\underset{n=0}{\sum ^{\mathrm{\infty }}}{\mathrm{P}}_r\left(n\right)\frac{x^n}{n!}=a_0\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{x^n}{n!}+a_1\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{x^n}{(n-1)!}$$$$+a_2\underset{n=2}{\sum ^{\mathrm{\infty }}}\frac{x^n}{(n-2)!}+....a_r\underset{n=r}{\sum ^{\mathrm{\infty }}}\frac{x^n}{(n-r)!}$$$$=a_0\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{x^n}{n!}+a_{1}x\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{x^{n-1}}{(n-1)!}+a_{2}x\underset{n=2}{\sum ^{\mathrm{\infty }}}\frac{x^{n-2}}{(n-2)!}+...$$$$...+a_r{x}^r\underset{n=r}{\sum ^{\mathrm{\infty }}}\frac{x^{n-r}}{(n-r)!}$$$$(a_0+a_{1}x+a_2{x}^2+...+a_r{x}^r)e^x$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com