n-Z-Find-the-coefficient-of-x-1-in-the-expansion-of-1-x-n-1-1-x-n-

Question Number 60406 by Tawa1 last updated on 20/May/19

n \in Z^{+}, Find the coefficient of x^{- 1} in the expansion of {(1 + x)}^{n} {(1 + \frac{1}{x})}^{n}

Answered by mr W last updated on 20/May/19

{(1 + x)}^{n} {(1 + \frac{1}{x})}^{n}

= \frac{1}{x^{n}} {(1 + x)}^{2 n}

= \frac{1}{x^{n}} \underset{k = 0}{\sum^{2 n}} C_{k}^{2 n} x^{k}

= \underset{k = 0}{\sum^{2 n}} C_{k}^{2 n} x^{k - n}

f o r k - n = - 1 \Rightarrow k = n - 1

c o e f . o f x^{- 1} i s C_{n - 1}^{2 n}

Commented by Tawa1 last updated on 20/May/19

God bless you sir, but the sum is confusing

Commented by Tawa1 last updated on 20/May/19

I now understand sir . God bless you sir

Commented by Tawa1 last updated on 20/May/19

But how can i sum this .

\overset{n}{} C_{0} \overset{n}{} C_{1} + \overset{n}{} C_{1} \overset{n}{} C_{2} + \overset{n}{} C_{2} \overset{n}{} C_{3} + \dots + \overset{n}{} C_{r} \overset{n}{} C_{r + 1}

Facebook Tweet Pin