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

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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.

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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 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

Medical Physics

Physics of Motion

Astrophysics

Translational Dynamics

Further Mechanics and Thermal Physics

Fluids

Study anywhere. Anytime. Across all devices.