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Chapter 10: Problem 13

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=e^{2 x}$$

Short Answer

Expert verified
Question: Express the Maclaurin series for the function \(f(x) = e^{2x}\) up to the fourth nonzero term, and determine its interval of convergence. Answer: The Maclaurin series for the function \(f(x) = e^{2x}\) up to the fourth nonzero term is: $$1 + 2x + 2x^2 + \frac{4}{3}x^3$$. The interval of convergence for this series is \(-\infty < x < \infty\), as the exponential function converges for all real values of x.

Step by step solution

01

Find the derivatives of the function

To find the first four nonzero terms of the Maclaurin series, we need to find the first four derivatives of the function \(f(x) = e^{2x}\). First derivative: $$f'(x) = \frac{d}{dx}e^{2x} = 2e^{2x}$$ Second derivative: $$f''(x) = \frac{d^2}{dx^2}e^{2x} = 4e^{2x}$$ Third derivative: $$f^{(3)}(x) = \frac{d^3}{dx^3}e^{2x} = 8e^{2x}$$ Fourth derivative: $$f^{(4)}(x) = \frac{d^4}{dx^4}e^{2x} = 16e^{2x}$$
02

Evaluate the derivatives at x = 0

Now that we have the derivatives, we need to evaluate them at \(x = 0\). $$f(0) = e^{2(0)} = 1$$ $$f'(0) = 2e^{2(0)} = 2$$ $$f''(0) = 4e^{2(0)} = 4$$ $$f^{(3)}(0) = 8e^{2(0)} = 8$$ $$f^{(4)}(0) = 16e^{2(0)} = 16$$
03

Write the first four nonzero terms of the Maclaurin series

Now, we can write the Maclaurin series up to the fourth nonzero term by replacing \(f^{(n)}(0)\) with the values we found in step 2 and plugging them into the Taylor series formula. $$f(x) \approx 1 + 2x + \frac{4}{2!}x^2 + \frac{8}{3!}x^3 = 1 + 2x + 2x^2 + \frac{4}{3}x^3$$
04

Write the power series using summation notation

We can express the general term of the Maclaurin series as: $$a_n = \frac{2^n}{n!}x^n$$ Now, we can write the power series using summation notation: $$f(x) = \sum_{n=0}^{\infty}\frac{2^n}{n!}x^n$$
05

Determine the interval of convergence

As the function is \(f(x) = e^{2x}\), and the exponential function converges for all real values of x, the interval of convergence for this Maclaurin series is: $$\text{Interval of convergence:} \, -\infty < x < \infty$$

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Most popular questions from this chapter

Write the Taylor series for \(f(x)=\ln (1+x)\) about 0 and find the interval of convergence. Evaluate \(f\left(-\frac{1}{2}\right)\) to find the value of \(\sum_{k=1}^{\infty} \frac{1}{k \cdot 2^{k}}.\)

Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$\frac{1}{\left(4+x^{2}\right)^{2}}$$

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{k+1}}{4^{k}}$$

Recall that the Taylor series for \(f(x)=1 /(1-x)\) about 0 is the geometric series \(\sum_{k=0}^{\infty} x^{k} .\) Show that this series can also be found as a case of the binomial series.

Find a power series that has (2,6) as an interval of convergence.
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