General Binomial Series

Question

Treating the binomial theorem for a negative index, the coefficient of x^3 in the series expansion of (1 - x)^{-2} valid for |x| < 1 is determined to be which value?

RankGuru · Accepted Answer

For a negative integer index the binomial theorem generalises to an infinite series whose coefficients follow the rising-factorial pattern, a simple application that JEE Advanced includes under binomial series. The expansion (1 - x)^{-2} = sum_{r=0}^{\infty} (r + 1) x^r, since (1 - x)^{-n} = sum_{r} C(n + r - 1, r) x^r and for n = 2 this gives C(r + 1, r) = r + 1. Therefore the coefficient of x^3 corresponds to r = 3, giving r + 1 = 4. Hence the required coefficient is 4. Option 3 mistakenly uses r instead of r + 1. Option 6 applies the formula for (1 - x)^{-3} whose coefficients are C(r + 2, 2). Option 1 treats the series as a simple geometric one with all unit coefficients, valid only for (1 - x)^{-1}. The convergence condition |x| < 1 ensures the infinite series is meaningful. Plausibility check: differentiating the geometric series 1/(1 - x) = sum x^r gives 1/(1 - x)^2 = sum (r + 1) x^r, and the x^3 coefficient is indeed 4, confirming the negative-index expansion.

General Binomial Series

Select the correct option:

🔒 Solution Hidden from View

More binomial theorem and its simple applications Practice Questions

Using the first-order binomial approximation for small quantities, the approximate value of (1.002)^...

When the trinomial (1 + x + x^2)^{3} is expanded completely into a polynomial in x, the coefficient ...

If the expansion of (1 + x)^n is split into terms with even index and terms with odd index, the sum ...

Within the expansion of (\sqrt{3} + \sqrt[3]{2})^{6}, the number of terms that turn out to be ration...

Suppose the coefficients of three consecutive terms in the expansion of (1 + x)^n are in the ratio 1...

By differentiating a standard expansion and substituting suitably, the weighted sum C(n,1) + 2 C(n,2...

Through integrating the binomial expansion over a unit interval, the alternating series C(n,0)/1 - C...

For the numerical expansion of (1 + x)^{n} with specific values applied, the greatest term is identi...

Using the binomial theorem to analyse remainders, the quantity 7^{83} is divided by 6 and the result...

When the very large number 3^{100} is computed and its final two digits are examined, the binomial t...

A question requires the coefficient of x^5 in the product expansion of (1 + x)^4 multiplied by (1 + ...

The combinatorial sum C(n,0) + C(n,1) + C(n,2) + ... + C(n,n), formed from a complete row of binomia...

About This Question

Solution