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Chapter 9: Problem 6

For what values of \(p\) does the Taylor series for \(f(x)=(1+x)^{p}\) centered at 0 terminate?

Short Answer

Expert verified
Answer: \(p = 1\)

Step by step solution

01

Find the first derivative of \(f(x)\)

To find the first derivative of \(f(x)\), we will use the power rule for differentiation. The power rule states that if \(f(x) = x^k\), then \(f'(x) = kx^{k-1}\). \(f(x) = (1+x)^{p}\) \(f'(x) = p(1+x)^{p-1}\) Now, let's evaluate the first derivative at 0. \(f'(0) = p(1+0)^{p-1} = p\)
02

Find the second derivative of \(f(x)\)

Now, we will find the second derivative of \(f(x)\) using the chain rule: \((u(v(x)))' = u'(v(x))\cdot v'(x)\). \(f'(x) = p(1+x)^{p-1}\) \(f''(x) = p(p-1)(1+x)^{p-2}\) Now, let's evaluate the second derivative at 0. \(f''(0) = p(p-1)(1+0)^{p-2} = p(p-1)\)
03

Analyze the derivatives at 0 and find the values of \(p\)

We have found two derivatives of \(f(x)\) at 0: \(f'(0) = p\) \(f''(0) = p(p-1)\) Now, we want to find the value of \(p\) for which there exists a natural number \(N\) such that \(f^N(0) = 0\) for all \(n > N\). For the first derivative \(f'(0)\), there is no such natural number because it only equals 0 if \(p=0\), and the power function is not well-defined for \(p=0\). For the second derivative \(f''(0)\), we can see that there is a natural number, namely \(N=1\), such that \(f^N(0) = 0\) for all \(n > N\). Therefore, the Taylor series for \(f(x)\) terminates when \(p = 1\).

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Most popular questions from this chapter

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. $$f(x)=b^{x}, \text { for } b > 0, b \neq 1$$

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{k}}{3^{k}}$$

Exponential function In Section 9.3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty< x <\infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=x^{2} e^{x}$$

Choose a Taylor series and center point to approximate the following quantities with an error of \(10^{-4}\) or less. $$\sqrt[3]{83}$$

Choose a Taylor series and center point to approximate the following quantities with an error of \(10^{-4}\) or less. $$\sin (0.98 \pi)$$
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