Integration Application

Question

Through integrating the binomial expansion over a unit interval, the alternating series C(n,0)/1 - C(n,1)/2 + C(n,2)/3 - ... reduces to which compact expression?

RankGuru · Accepted Answer

Series in which binomial coefficients are divided by successive integers are evaluated by integrating the expansion term by term, a technique that complements differentiation in JEE Advanced. Consider (1 - x)^n = sum_{r=0}^{n} C(n, r)(-1)^r x^r and integrate both sides from 0 to 1. The left integral is integral_0^1 (1 - x)^n dx = [-(1 - x)^{n+1}/(n+1)]_0^1 = 1/(n+1). The right side integrates term by term to sum_{r=0}^{n} C(n, r)(-1)^r /(r+1), which is exactly C(n,0)/1 - C(n,1)/2 + C(n,2)/3 - ... Hence the series equals 1/(n+1). Option 1/n drops the plus one that arises from raising the power during integration. Option 1/(n-1) mishandles the limits. Option 2^n/(n+1) wrongly multiplies by a coefficient sum. The integration limits 0 to 1 are chosen precisely to produce the reciprocal integer denominators, because integrating x^r over this interval yields exactly 1/(r+1). The legitimacy of integrating the finite sum term by term is guaranteed since a polynomial is a finite combination of continuous functions, so no convergence subtlety arises. Plausibility check: for n = 1 the series is 1 - 1/2 = 1/2 = 1/(1 + 1), and for n = 2 it is 1 - 1 + 1/3 = 1/3 = 1/(2 + 1), confirming the formula across small cases.

Integration Application

Select the correct option:

🔒 Solution Hidden from View

More binomial theorem and its simple applications Practice Questions

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

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

Treating the binomial theorem for a negative index, the coefficient of x^3 in the series expansion o...

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...

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

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

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 + ...

Evaluating the alternating combination C(n,0) - C(n,1) + C(n,2) - C(n,3) + ... over a full binomial ...

About This Question

Solution