Chapter 1: Q3P (page 29)

Show that if p is a positive integer, then $(\begin{array}{l} p \\ n \end{array}) = 0$ when $n > p$ , so $(1 + x)^{p} = \sum (\begin{array}{l} p \\ n \end{array}) x^{n}$ is just a sum of $p + 1$ terms, from $n = 0$ to $n = p$ . For example, $(1 + x)^{2}$ has $3$ terms, $(1 + x)^{3}$ has $4$ terms, etc. This is just the familiar binomial theorem.

Short Answer

Expert verified

The statement has been proven.

Step by step solution

Given Information

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

localid="1657347784194" $(\begin{array}{l} p \\ n \end{array}) = \frac{p (p - 1) (p - 2) \dots (p - p) \dots (p - n + 1)}{n!}$

localid="1657347855207" $\frac{p (p - 1) (p - 2) \dots (p - p) \dots (p - n + 1)}{n!} = 0$

localid="1657347906098" $\frac{p!}{p! (p - p)!} = 1$

localid="1657347974064" $(\begin{array}{l} p \\ n \end{array}) \neq 0$ only for localid="1657348003107" $n \leq p$ andlocalid="1657348034900" $n \geq 0$

Solve further.

localid="1657348111696" $(1 + x)^{p} = \sum_{n = 0}^{\infty} (\begin{array}{l} p \\ n \end{array}) x^{n}$

localid="1657348136816" $(1 + x)^{p} = (\begin{array}{l} p \\ 0 \end{array}) + (\begin{array}{l} p \\ 1 \end{array}) x + (\begin{array}{l} p \\ 2 \end{array}) x^{2} + (\begin{array}{l} p \\ 3 \end{array}) x^{3} + \dots + (\begin{array}{l} p \\ p \end{array}) x^{p}$

The expansion haslocalid="1657348245474" $p + 1$ terms for localid="1657348169040" $n = 0 - p$
.

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.

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

Magnetism and Electromagnetic Induction

Modern Physics

Medical Physics

Turning Points in Physics

Linear Momentum

Fields in Physics

Study anywhere. Anytime. Across all devices.

Show that if p is a positive integer, thenpn=0when n>p , so (1+x)p=∑pnxnis just a sum ofp+1terms, from n=0 to n=p . For example,(1+x)2has 3 terms,(1+x)3 has4terms, etc. This is just the familiar binomial theorem.

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

Magnetism and Electromagnetic Induction

Modern Physics

Medical Physics

Turning Points in Physics

Linear Momentum

Fields in Physics

Study anywhere. Anytime. Across all devices.