Evaluate-1-2-3-1-3-3-2-4-3-3-by-considering-the-series-expansion-of-an-expression-of-the-form-P-x-e-x-where-P-x-is-a-suitably-chosen-polynomial-in-x-

Question Number 2157 by Yozzi last updated on 05/Nov/15

E v a l u a t e

1 + \frac{2^{3}}{1!} + \frac{3^{3}}{2!} + \frac{4^{3}}{3!} + \dots

b y c o n s i d e r i n g t h e s e r i e s e x p a n s i o n

o f a n e x p r e s s i o n o f t h e f o r m P (x) e^{x}

w h e r e P (x) i s a s u i t a b l y c h o s e n

p o l y n o m i a l i n x .

Commented by RasheedAhmad last updated on 15/Nov/15

1 + \frac{2^{3}}{1!} + \frac{3^{3}}{2!} + \frac{4^{3}}{3!} + \dots

= \frac{1^{3}}{0!} + \frac{2^{3}}{1!} + \frac{3^{3}}{2!} + \frac{4^{3}}{3!} + \dots

= \frac{1^{3} . 1}{0! . 1} + \frac{2^{3} . 2}{1! . 2} + \frac{3^{3} . 3}{2! . 3} + \frac{4^{3} . 4}{3! . 4} + \dots

= \frac{1^{4}}{1!} + \frac{2^{4}}{2!} + \frac{3^{4}}{3!} + \frac{4^{4}}{4!} + \dots + \frac{n^{4}}{n!} + \dots