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

n ∈ Z^+ , Find the coefficient of x^(−1) in the expansion of (1 + x)^n (1 + (1/x))^n

$$\mathrm{n}\:\in\:\mathbb{Z}^{+} ,\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\:\mathrm{x}^{−\mathrm{1}} \:\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\:\:\left(\mathrm{1}\:+\:\mathrm{x}\right)^{\mathrm{n}} \left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{n}} \\ $$

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

(1+x)^n (1+(1/x))^n =(1/x^n )(1+x)^(2n) =(1/x^n )Σ_(k=0) ^(2n) C_k ^(2n) x^k =Σ_(k=0) ^(2n) C_k ^(2n) x^(k−n) for k−n=−1⇒k=n−1 coef. of x^(−1) is C_(n−1) ^(2n)

$$\left(\mathrm{1}+{x}\right)^{{n}} \left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{{n}} \\ $$$$=\frac{\mathrm{1}}{{x}^{{n}} }\left(\mathrm{1}+{x}\right)^{\mathrm{2}{n}} \\ $$$$=\frac{\mathrm{1}}{{x}^{{n}} }\underset{{k}=\mathrm{0}} {\overset{\mathrm{2}{n}} {\sum}}{C}_{{k}} ^{\mathrm{2}{n}} {x}^{{k}} \\ $$$$=\underset{{k}=\mathrm{0}} {\overset{\mathrm{2}{n}} {\sum}}{C}_{{k}} ^{\mathrm{2}{n}} {x}^{{k}−{n}} \\ $$$${for}\:{k}−{n}=−\mathrm{1}\Rightarrow{k}={n}−\mathrm{1} \\ $$$${coef}.\:{of}\:{x}^{−\mathrm{1}} \:{is}\:{C}_{{n}−\mathrm{1}} ^{\mathrm{2}{n}} \\ $$

Commented by Tawa1 last updated on 20/May/19

God bless you sir, but the sum is confusing

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir},\:\:\mathrm{but}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{is}\:\mathrm{confusing} \\ $$

Commented by Tawa1 last updated on 20/May/19

I now understand sir. God bless you sir

$$\mathrm{I}\:\mathrm{now}\:\mathrm{understand}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Commented by Tawa1 last updated on 20/May/19

But how can i sum this. ^n C_0 ^n C_1 + ^n C_1 ^n C_2 + ^n C_2 ^n C_3 + ... + ^n C_r ^n C_(r + 1)

$$\mathrm{But}\:\mathrm{how}\:\mathrm{can}\:\mathrm{i}\:\mathrm{sum}\:\mathrm{this}.\:\: \\ $$$$\:\:\:\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{0}} \overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{1}} \:+\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{1}} \overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{2}} \:+\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{2}} \overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{3}} \:+\:...\:+\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{r}} \overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{r}\:+\:\mathrm{1}} \\ $$

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