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Chapter 10: Problem 34

Apply Taylor's Theorem to find the binomial series (centered at \(c=0\) ) for the function, and find the radius of convergence. \(f(x)=\sqrt{1+x}\)

Short Answer

Expert verified
The binomial series for the function \(f(x)=\sqrt{1+x}\) (centered at \(c=0\)) is \(1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - ...\) and the radius of convergence is 1.

Step by step solution

01

Identify the Form of the Given Function

The given function, \(f(x)=\sqrt{1+x}\), can be written in the form \((1+x)^p\) where \(p=1/2\).
02

Binomial Series Formula

The general formula for the binomial series is \((1+x)^p = 1 + px+ \frac{p(p-1)x^2}{2!} + \frac{p(p-1)(p-2)x^3}{3!} + ...\) .
03

Apply Binomial Series Formula

Substitute \(p=1/2\) into the binomial series formula to get the expansion for the given function: \(f(x) = \sqrt{1+x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - ...\)
04

Compute the Radius of Convergence

The radius of convergence for the binomial series is 1 when \(|x|<1\). This means it is valid where \(|x|<1\), hence the radius of convergence is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Binomial Series
The binomial series is an infinite series expansion of expressions raised to any real power. It's a way of expressing functions like \((1+x)^p\) where \(p\) can be a fraction, a negative number, or any real number, in series form. This is particularly useful because it allows us to approximate these expressions using polynomials, which are much simpler to work with.

A binomial series consists of terms arranged based on increasing powers of \(x\):
  • The first term is always 1.
  • The second term is \(px\), where \(p\) is the exponent from the original expression.
  • The third term, and those following, take into account factors of the exponent decrementing by one each time. For instance, the third term includes \(\frac{p(p-1)x^2}{2!}\).
These terms are derived step by step and combined to create the overall series. Understanding this concept is key in utilizing the series effectively in problems involving approximation and calculus.
The Concept of Radius of Convergence
The radius of convergence is a crucial concept in series expansion, especially when dealing with power series like the binomial series.

In essence, the radius of convergence defines the interval within which the series converges to a real number. Beyond this interval, the series may not necessarily provide valid results. Testing the interval involves checking the absolute values for \(x\) where the expansion remains reliable.
  • For our function \(f(x)=\sqrt{1+x}\), the binomial series only converges when \(|x| < 1\).
  • Hence, the radius of convergence here is 1, representing the boundary within which the series accurately approximates the original function.
This conceptual understanding assures students that they can use the series within this radius without losing validity, allowing for effective approximation of functions.
The Binomial Series Formula
The binomial series formula provides a structured approach to expanding expressions of the form \((1+x)^p\).

Its general representation is:
  • \((1+x)^p = 1 + px + \frac{p(p-1)x^2}{2!} + \frac{p(p-1)(p-2)x^3}{3!} + ...\)
  • This continues with each subsequent term involving additional decrements in \(p\), multiplied by higher powers of \(x\) divided by their respective factorial.
To apply the formula:
  • Plug in the specific \(p\) value from the problem into the formula.
  • Carry out the multiplications and divisions as per the structure given, to arrive at a series.
For example, for \(f(x) = \sqrt{1+x}\) where \(p = \frac{1}{2}\), this results in the expanded series: \[ f(x) = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - ... \]This step-by-step expansion forms the basis for solving approximation problems using the binomial series.

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