Sum-the-series-n-0-P-r-n-x-n-n-where-P-r-n-is-a-polynomial-of-degree-r-in-n-

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

Sum the series \underset{n = 0}{\sum^{\infty}} P_{r} (n) \frac{x^{n}}{n!}, where P_{r} (n)

is a polynomial of degree r in n .

Answered by user1 last updated on 31/Oct/14

P_{r} (n) = a_{0} + a_{1} n + a_{2} n (n - 1) + \dots . + a_{r} n (n - 1) \dots (n - r + 1),

\underset{n = 0}{\sum^{\infty}} P_{r} (n) \frac{x^{n}}{n!} = a_{0} \underset{n = 0}{\sum^{\infty}} \frac{x^{n}}{n!} + a_{1} \underset{n = 1}{\sum^{\infty}} \frac{x^{n}}{(n - 1)!}

+ a_{2} \underset{n = 2}{\sum^{\infty}} \frac{x^{n}}{(n - 2)!} + \dots . a_{r} \underset{n = r}{\sum^{\infty}} \frac{x^{n}}{(n - r)!}

= a_{0} \underset{n = 0}{\sum^{\infty}} \frac{x^{n}}{n!} + a_{1} x \underset{n = 1}{\sum^{\infty}} \frac{x^{n - 1}}{(n - 1)!} + a_{2} x \underset{n = 2}{\sum^{\infty}} \frac{x^{n - 2}}{(n - 2)!} + \dots

\dots + a_{r} x^{r} \underset{n = r}{\sum^{\infty}} \frac{x^{n - r}}{(n - r)!}

(a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{r} x^{r}) e^{x}

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