Chapter 1: Q3P (page 35)

Show that if $p$ is a positive integer, then $(\frac{p}{n}) = 0$ 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

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

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

role="math" localid="1657435976526" $\frac{p (p - 1) (p - 2) \dots (p - p) \dots (p - n + 1)}{n!} = 0 \frac{p!}{p! (p - p)!} = 1 (\begin{array}{l} p \\ n \end{array}) \neq 0 only for n \leq p and n \geq 0$

Solve further.

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

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

The expansion has $p + 1$ terms for $n = 0 - p$ .

The statement has been proven.

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Short Answer

Step by step solution

Given Information

Definition of the binomial series.

Prove the statement.

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Show that if pis a positive integer, then (pn)=0 when n>p,so (1+x)p=∑(pn)xnis just a sum of p+1terms, from n=0to n=p. For example, (1+x)2has 3terms, (1+x)3has 4terms, 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

Further Mechanics and Thermal Physics

Space Physics

Linear Momentum

Measurements

Geometrical and Physical Optics

Rotational Dynamics

Study anywhere. Anytime. Across all devices.