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Chapter 7: Problem 75

Write the power series for \((1+x)^{k}\) in terms of binomial coefficients.

Short Answer

Expert verified
\[(1+x)^{k} = \binom{k}{0}x^0 + \binom{k}{1}x^1 + \binom{k}{2}x^2 +...+ \binom{k}{k}x^k\]

Step by step solution

01

Declaring the Binomial Theorem

The Binomial theorem is a mathematical formula that describes the algebraic expansion of powers of a binomial. It is represented as: \[(1+x)^{k} = \sum_{n=0}^{k} \binom{k}{n} x^n\] where \(k\) is any real number and \(\binom{k}{n}\) represents the binomial coefficient.
02

Understanding Binomial Coefficients

The binomial coefficient, \(\binom{k}{n}\), is the number of ways in which you can choose \(n\) elements from a pool of \(k\) elements. It can be computed using the formula: \[\binom{k}{n} = \frac{k!}{n!(k-n)!}\] where \(k!\) denotes the factorial of \(k\), which means the product of all positive integers less than or equal to \(k\).
03

Applying the Binomial Theorem

By the Binomial theorem, the power series for \((1+x)^{k}\) in terms of binomial coefficients is: \[(1+x)^{k} = \binom{k}{0}x^0 + \binom{k}{1}x^1 + \binom{k}{2}x^2 +...+ \binom{k}{k}x^k\] Each term of the power series corresponds to a binomial coefficient. The power on \(x\) in each term also corresponds to the number of ways to choose elements from a pool of \(k\) elements.

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