Chapter 8: Q 29 (page 704)

Use the Maclaurin series for $e^{x}$ to find power series representations
for $g (x) = x^{2} e^{x}, g^{'} (x)$ and $\int g (x) d x$

Short Answer

Expert verified

The power series representation for $g (x) = x^{2} e^{x}$ is $g (x) = \sum_{k = 0}^{\infty} \frac{1}{k!} x^{k + 2}$

Step by step solution

Given information

The function is $f (x) = e^{x}$

Find the Maclaurin series for the function

The Maclaurin series for the function $f (x) = e^{x}$ is

$1 + x + \frac{1}{2!} x^{2} + \frac{1}{3!} x^{3} + \frac{1}{4!} x^{4} + \dots$

Or $f (x) = \sum_{k = 0}^{\infty} \frac{1}{k!} x^{k}$

Hence, $g (x) = x^{2} e^{x}$

$g (x) = x^{2} \sum_{k = 0}^{8} \frac{1}{k!} x^{k} = \sum_{k = 0}^{8} \frac{1}{k!} x^{k + 2}$

Find the power series representation for g(x)=x2ex

The power series representation for $g (x) = x^{2} e^{x}$ is

$g (x) = x^{2} \sum_{k = 0}^{\infty} \frac{1}{k!} x^{k}$

$g (x) = \sum_{k = 0}^{\infty} \frac{1}{k!} x^{k + 2}$

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Use the Maclaurin series for $e^{x}$ to find power series representations
for $g (x) = x^{2} e^{x}, g^{'} (x)$ and $\int g (x) d x$

Short Answer

Step by step solution

Given information

Find the Maclaurin series for the function

Find the power series representation for g(x)=x2ex

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Use the Maclaurin series for exto find power series representationsfor g(x)=x2ex,g'(x)and ∫g(x)dx

Short Answer

Step by step solution

Given information

Find the Maclaurin series for the function

Find the power series representation for g(x)=x2ex

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Use the Maclaurin series for $e^{x}$ to find power series representations
for $g (x) = x^{2} e^{x}, g^{'} (x)$ and $\int g (x) d x$