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.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Solid State Physics

Scientific Method Physics

Fluids

Waves Physics

Oscillations

Study anywhere. Anytime. Across all devices.

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

Force

Solid State Physics

Scientific Method Physics

Fluids

Waves Physics

Oscillations

Study anywhere. Anytime. Across all devices.

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