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Chapter 10: Problem 8

Write the Maclaurin series for \(e^{2 x}\)

Short Answer

Expert verified
Answer: The Maclaurin series for the function \(e^{2x}\) is given by \(e^{2x} = \sum_{n=0}^{\infty} \frac{2^n}{n!} x^n\).

Step by step solution

01

Find the general form for the nth derivative

For a function of the form \(e^{ax}\), the derivative would always be a multiple of the function itself. In this case, since our function is \(e^{2x}\), the nth derivative would be \((2^n)e^{2x}\).
02

Calculate the nth Maclaurin coefficient

By definition, the nth coefficient of the Maclaurin series is given by the expression: $$a_n = \frac{f^{(n)}(0)}{n!}$$ Here, \(f^{(n)}(0)\) represents the value of the nth derivative of the function at x=0. In our case, since the nth derivative is \((2^n)e^{2x}\), we have: $$f^{(n)}(0) = (2^n)e^{2\cdot 0} = (2^n)e^0 = 2^n$$ So, the nth Maclaurin coefficient for our function is: $$a_n = \frac{2^n}{n!}$$
03

Write the general term in the Maclaurin series

The Maclaurin series is represented as the sum of the terms of the sequence of coefficients multiplied by the corresponding power of x: $$\sum_{n=0}^{\infty} a_n x^n$$ From step 2, since \(a_n = \frac{2^n}{n!}\): $$\sum_{n=0}^{\infty} \frac{2^n}{n!} x^n$$
04

Present the Maclaurin series for \(e^{2x}\)

Based on our calculations in step 3, the Maclaurin series for the function \(e^{2x}\) can be written as: $$e^{2x} = \sum_{n=0}^{\infty} \frac{2^n}{n!} x^n$$

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Most popular questions from this chapter

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}\). $$\int_{0}^{0.2} \frac{\ln (1+t)}{t} d t$$

Consider the following common approximations when \(x\) is near zero. a. Estimate \(f(0.1)\) and give the maximum error in the approximation. b. Estimate \(f(0.2)\) and give the maximum error in the approximation. $$f(x)=\tan ^{-1} x \approx x$$

By comparing the first four terms, show that the Maclaurin series for \(\cos ^{2} x\) can be found (a) by squaring the Maclaurin series for \(\cos x,\) (b) by using the identity \(\cos ^{2} x=(1+\cos 2 x) / 2,\) or \((\mathrm{c})\) by computing the coefficients using the definition.

Find the next two terms of the following Taylor series. $$\sqrt{1+x}: 1+\frac{1}{2} x-\frac{1}{2 \cdot 4} x^{2}+\frac{1 \cdot 3}{2 \cdot 4 \cdot 6} x^{3}-\cdots$$

Identify the functions represented by the following power series. $$\sum_{k=2}^{\infty} \frac{k(k-1) x^{k}}{3^{k}}$$
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