Chapter 1: Q1. (page 29)

Show that the binomial coefficients $(\begin{array}{l} - 1 \\ n \end{array}) = (- 1)^{n}$

Short Answer

Expert verified

The statement has been proven.

Step by step solution

Given Information

The binomial series

Definition of the binomial series

The Taylor series for the function given by is the binomial series, where is an arbitrary complex number.

Prove the statement

The binomial series states that $(1 + x)^{p} = \sum_{n = 0}^{\infty} (\begin{array}{l} p \\ n \end{array}) x^{n}$

The formula states that $(\begin{array}{l} p \\ n \end{array}) = \frac{p (p - 1) (p - 2) \dots (p - n + 1)}{n!}$

$(1 + x)^{- 1} = \sum_{n = 0}^{\infty} (\begin{matrix} - 1 \\ n \end{matrix}) x^{n}$

$= 1 - x + \frac{(- 1) (- 2)}{2!} x^{2} + \frac{(- 1) (- 2) (- 3)}{3!} x^{3} + \dots$

$= 1 - x + x^{2} - x^{3} + \dots$

$= \sum_{n = 0}^{\infty} (- 1)^{n} x^{n}$

${Hence}_{α,} \sum_{n = 0}^{\infty} (\begin{matrix} - 1 \\ n \end{matrix}) x^{n} = \sum_{n = 0}^{\infty} (- 1)^{n} x^{n} .$

The statement has been proven.

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Show that the binomial coefficients $(\begin{array}{l} - 1 \\ n \end{array}) = (- 1)^{n}$

Short Answer

Step by step solution

Given Information

Definition of the binomial series

Prove the statement

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Show that the binomial coefficients-1n=(-1)n

Short Answer

Step by step solution

Given Information

Definition of the binomial series

Prove the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Show that the binomial coefficients $(\begin{array}{l} - 1 \\ n \end{array}) = (- 1)^{n}$