Prove that Σ_(i=0) ^(n+1) (((n+1)),(i) ) i^n (−1)^(i+1) =0 for ∀n∈N.


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Question Number 7861 by Yozzia last updated on 21/Sep/16

$P r o v e t h a t \underset{i = 0}{\sum^{n + 1}} (\begin{matrix} n + 1 \\ i \end{matrix}) i^{n} {(- 1)}^{i + 1} = 0$ $f o r \forall n \in N .$
Commented by FilupSmith last updated on 22/Sep/16

$a t t e m p t$ $\underset{i = 0}{\sum^{n + 1}} (\begin{matrix} n + 1 \\ i \end{matrix}) = \underset{i = 0}{\sum^{k}} \frac{k!}{i! \cdot (k - i)!} = k! \underset{i = 0}{\sum^{k}} \frac{1}{i! \cdot (k - i)!}$ $n + 1 = k \Rightarrow n = k - 1$ $\underset{i = 0}{\sum^{k}} S = k! (\frac{1}{0! \times k!} + \frac{1}{1! \times (k - 1)!} + \frac{1}{2! \times (k - 2)!} + . . . + \frac{1}{(k - 1)! \times 1!} + \frac{1}{k! \times 0!})$ $∴ \underset{i = 0}{\sum^{n + 1}} (\begin{matrix} n + 1 \\ i \end{matrix}) i^{n} {(- 1)}^{n + 1} = \underset{i = 0}{\sum^{k}} {S i}^{k - 1} {(- 1)}^{k}$ $\underset{i = 0}{\sum^{k}} {S i}^{k - 1} = k! (\frac{0^{k - 1}}{0! \times k!} + \frac{1^{k - 1}}{1! \times (k - 1)!} + \frac{2^{k - 1}}{2! \times (k - 2)!} + . . . + \frac{{(k - 1)}^{k - 1}}{(k - 1)! \times 1!} + \frac{k^{k - 1}}{k! \times 0!})$ $\underset{i = 0}{\sum^{k}} {S i}^{k - 1} = \frac{0^{k - 1}}{0!} + \frac{k \times 1^{k - 1}}{1!} + \frac{k (k - 1) 2^{k - 1}}{2!} + \frac{k (k - 1) (k - 2) 3^{k - 1}}{3!} + . . . + \frac{k {(k - 1)}^{k - 1}}{1!} + \frac{k^{k - 1}}{0!}$ $∴ \underset{i = 0}{\sum^{k}} {S i}^{k - 1} {(- 1)}^{k} = \frac{0^{k - 1}}{0!} + \frac{k \times 1^{k - 1}}{1!} - \frac{k (k - 1) 2^{k - 1}}{2!} + \frac{k (k - 1) (k - 2) 3^{k - 1}}{3!} - . . . + \frac{k {(k - 1)}^{k - 1}}{1!} - \frac{k^{k - 1}}{0!}$ $note : + if previous denominator has a even term$ $∴ = \frac{0^{k - 1}}{0!} + (\frac{k \times 1^{k - 1}}{1!} + \frac{k (k - 1) (k - 2) 3^{k - 1}}{3!} + . . .) - (\frac{k (k - 1) 2^{k - 1}}{2!} + \frac{k (k - 1) (k - 2) (k - 3) 4^{k - 1}}{4!} + . . .)$ $w o r k i n g$ $c u r r e n t l y u n s u r e h o w t o c o n t i n u e$
Commented by prakash jain last updated on 01/Oct/16

$f (x) = {(1 - e^{x})}^{n + 1}$ $= - \underset{i = 0}{\sum^{n + 1}}^{n + 1} C_{i} e^{i x} {(- 1)}^{i + 1}$ $f^{n} (x) = - \underset{i = 0}{\sum^{n}}^{n + 1} C_{i} i^{n} e^{i x} {(- 1)}^{i + 1} = required expression$ $at x = 0$ $Also f (x) = {(1 - e^{x})}^{n + 1} = L H S$ $f^{'} (x) = - (n + 1) {(1 - e^{x})}^{n} (- e^{x})$ $f^{2} (x) = - [(n + 1) (e^{x}) {(1 - e^{x})}^{n} - (n + 1) {n e}^{x} {(1 - e^{x})}^{n - 1} (- e^{x})]$ $⋮$ $f^{n} (0) = 0 [∵ (1 - e^{x}) is factor till n^{t h} d i f f e r e n t i a t i o n]$ $hence$ $- \underset{i = 0}{\sum^{n}}^{n + 1} C_{i} i^{n} {(- 1)}^{i + 1}$ $please check$
Commented by Yozzia last updated on 30/Sep/16

$T h a t s e e m s s u f f i c i e n t t o s h o w t h a t$ $t h e e q u a t i o n i s t r u e . N i c e l y d o n e!$ $I l i k e t h e a p p l i c a t i o n o f t h e e x p o n e n t i a l$ $f u n c t i o n a n d b i n o m i a l t h e o r e m .$
	Answered by prakash jain last updated on 22/Sep/16

	$n = 2$ $-^{3} C_{0} . 0 + 1^{2}^{3} C_{1} - 2^{2}^{3} C_{2} + 3^{2}^{3} C_{3}$ $= - 0 + 3 - 12 + 9 = 0$ $n = 3$ $- 0 + 1^{3} \times^{4} C_{1} - 2^{3} \times^{4} C_{2} + 3^{3} \times^{4} C_{3} - 4^{3} \times^{4} C_{4}$ $= 1 \times 4 - 8 \times 6 + 27 \times 4 - 64$ $= 4 - 48 - 108 - 64 = 0$
	Commented by Yozzia last updated on 22/Sep/16

	$(\begin{matrix} 3 \\ 0 \end{matrix}) h a s c o e f f i c i e n t 0 f o r T (n) = (\begin{matrix} n + 1 \\ i \end{matrix}) i^{n} {(- 1)}^{n + 1} n ⩾ 0 .$
	Commented by Yozzia last updated on 22/Sep/16
	$I met this series in proving by induction that the general coefficient a_n of the terms in the power series of the W−Lambert function is a_n =(((−1)^(n+1) n^(n−2) )/((n−1)!)), n≥1 W(x)=Σ_(n=1) ^∞ (((−1)^(n+1) n^(n−2) x^n )/((n−1)!)) {−1≤x≤1} W(x)=f^(−1) (xe^x ) for −1≤x≤1.$
	$I m e t t h i s s e r i e s i n p r o v i n g b y i n d u c t i o n t h a t$ $t h e g e n e r a l c o e f f i c i e n t a_{n} o f t h e t e r m s i n t h e p o w e r s e r i e s$ $o f t h e W - L a m b e r t f u n c t i o n i s$ $a_{n} = \frac{{(- 1)}^{n + 1} n^{n - 2}}{(n - 1)!}, n ⩾ 1$ $W (x) = \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n + 1} n^{n - 2} x^{n}}{(n - 1)!} {- 1 ⩽ x ⩽ 1}$ $W (x) = f^{- 1} ({x e}^{x}) f o r - 1 ⩽ x ⩽ 1 .$
	Commented by prakash jain last updated on 22/Sep/16

	$Ok my mistake will look at again .$
	Commented by FilupSmith last updated on 22/Sep/16

	$This is too complicated . I {don}^{'} t know$ $much about the W - F u n c t i o n$
	Commented by prakash jain last updated on 22/Sep/16

	$n = 3 n o t z e r o .$
	Commented by Yozzia last updated on 22/Sep/16

	$C h e c k t h a t 3^{3} t e r m .$
	Commented by prakash jain last updated on 30/Sep/16

	$Please check answer in comment .$
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