The expansion 1-xn - 1 - nx - nn-1-2-x2- is valid if -x- - 1-

It is valid only if |x| lt; 1 The expansion (1+x)^n = 1 + nx + n(n 1)/2!(x)^2......... has infinite number of terms. If |x| gt; 1, the value on the R. ...

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Standard XI

Mathematics

Binomial Theorem for Any Index

The expansion...

Question

The expansion ${(1 + x)}^{n}$ = 1 + nx + $\frac{n (n - 1)}{2!}$ ${(x)}^{2}$ ........... is valid if |x| > 1.

True

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False

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Solution

The correct option is B
False

It is valid only if |x| < 1

The expansion ${(1 + x)}^{n}$ = 1 + nx + $\frac{n (n - 1)}{2!}$ ${(x)}^{2}$ ......... has infinite number of terms. If |x| > 1, the value on the R.H.S might become infinite. L.H.S. will always be finite if n is finite. E.g.: consider ${(1 + 1)}^{n}$ = 1 + n + $\frac{n (n - 1)}{2!}$ ........ L.H.S. is 2n but R.H.S. will be infinite. So we want |x| < 1.

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Similar questions

Q. If x₁,x₂,x₃,x₄….x_2n+1are in Arithmetic Progression, then find the value of [(x_2n+1–x₁)/ (x_2n+1+x₁)] + [(x_2n–x₂)/ (x_2n–x₂)]………….. [(x_n+2–x_n)/ (x_n+2–x_n)]

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Q. If

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, then the coefficient of

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in the expansion

{(1 + x + x^{2} + x^{3} + . . .)}^{2}

Q. If

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then the coefficient of

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Q. Let

(1 + x)^{n} = 1 + n x + \frac{n (n - 1) x^{2}}{2} + \dots | x | < 1, n ϵ Z

Then the sum of the series

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The expansion (1+x)n = 1 + nx + n(n−1)2!(x)2........... is valid if |x| > 1.

The expansion ${(1 + x)}^{n}$ = 1 + nx + $\frac{n (n - 1)}{2!}$ ${(x)}^{2}$ ........... is valid if |x| > 1.