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# please tell me about combination and binomial theorem

## 1 Answers

11 years ago

Hi,

The binomial theorem, for n a natural number,

(a + b)^n = a^n + n a^(n-1) b + n (n-1) a^(n-2) b^2 / 2 + ... + (n! / ((n - i)! i!) a^(n - i) b^i + ... + b^n.

may be proven by Mathematical Induction.

If we set a equal to one and replace b by x, we obtain the power series

(1 + x) ^ n = 1 + n x + n (n - 1) x^2 / 2 + n (n - 1) (n - 2) x^3 / 6 + ... (n! / ((n - i)! i!) x^i + ... + x^n.

By additional Mathematical Induction, it may be shown that this binomial expansion holds for any rational n; however, the series becomes infinite, with a radius of convergence abs(x) < 1. Because of the uniqueness of derivatives, this series is the Taylor series.

For example, take n = 1 / 2 to obtain the Taylor series for a square root

sqrt(1 + x) = (1 + x) ^ (1 / 2) = 1 + (1 / 2) x - (1 / 8) x ^ 2 + (1 / 16) x ^ 3 + ... + ((- 1) ^ i (i - 3 / 2)! / ((- 3 / 2)! i!)) x ^ i + ....

The general term has been written in terms of the factorial of a half-integer. It may be evaluated by means of the gamma function. This series converges for any abs(x) < 1, it does so practically fast only when abs(x) < = 1 / 2. Also, it may be expeditious to split the real and imaginary components.

The ratio of the i-th term to the (i-1)-th term is - (i - 3 / 2) x / i. There in neither the gamma nor the factorial function in this ratio.

Please feel free to post as many doubts on our disucssion forum as you can. If you find any question difficult to understand - post it here and we will get you the answer and detailed solution very quickly. We are all IITians and here to help you in  your IIT JEE preparation.

All the best and wish u a bright future!!!

Regards,
Askiitians Expert
Amit - IT BHU

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