prove-that-k-0-k-2-2-x-k-k-3-e-x-x-3-x-2-x-1-x-2-2-2x-3-

Question Number 88723 by M±th+et£s last updated on 12/Apr/20

p r o v e t h a t

\underset{k = 0}{\sum^{\infty}} \frac{{(k + 2)}^{2} x^{k}}{(k + 3)!} = \frac{e^{x}}{x^{3}} (x^{2} - x + 1) - \frac{x^{2} + 2}{2 x^{3}}

Answered by mind is power last updated on 12/Apr/20

{(k + 2)}^{2} = (k + 3) (k + 2) - (k + 3) + 1

Σ \frac{x^{k}}{(k + 1)!} - Σ \frac{x^{k}}{(k + 2)!} + Σ \frac{x^{k}}{(k + 3)!}

\frac{1}{x} \sum_{k ⩾ 0} \frac{x^{k + 1}}{(k + 1)!} - \frac{1}{x^{2}} Σ \frac{x^{k + 2}}{(k + 2)!} + \frac{1}{x^{3}} Σ \frac{x^{k + 3}}{(k + 3)!}

= \frac{1}{x} (e^{x} - 1) - \frac{1}{x^{2}} (e^{x} - 1 - x) + \frac{1}{x^{3}} (e^{x} - 1 - x - \frac{x^{2}}{2})

= e^{x} (\frac{1}{x} - \frac{1}{x^{2}} + \frac{1}{x^{3}}) - \frac{1}{x^{3}} - \frac{1}{2 x^{}}

= \frac{e^{x}}{x^{3}} (1 - x + x^{2}) - \frac{2 + x^{2}}{2 x^{3}}

Commented by M±th+et£s last updated on 12/Apr/20

t h a n k y o u s i r .

y o u c a n t a k e a l o o k o n Q 88197

Commented by mind is power last updated on 12/Apr/20

y e a h w i t h e p l e a s u r

Commented by I want to learn more last updated on 12/Apr/20

Sir, please help with Q 88731