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please tell me about combination and binomial theorem

please tell me about combination and binomial theorem


 

Grade:11

1 Answers

Amit Askiitiansexpert
112 Points
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.


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Regards,
Askiitians Expert
Amit - IT BHU

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