Chapter 8: Q 18. (page 704)

Maclaurin and Taylor series: Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series. $e^{x}, x_{0} = 1$

Short Answer

Expert verified

$e + e (x - 1) + \frac{e}{2!} (x - 1)^{2} + \frac{e}{3!} (x - 1)^{3} + \frac{e}{4!} (x - 1)^{4} + \dots$ or $P_{n} (x) = \sum_{k = 0}^{\infty} \frac{e}{k!} (x - 1)^{k}$

Step by step solution

Given information

$e^{x}, x_{0} = 1$

Concept

The formula used: $P_{n} (x) = \sum_{k = 0}^{\infty} \frac{f^{k} (x_{0})}{k!} {(x - x_{0})}^{n}$

Calculation

Consider the function $f (x) = e^{x}$

Since for any function $f$ with a derivative of order $n$ the Taylor series at $x = 1$ is given by

P_{n} (x) = f (1) + f^{'} (1) (x - 1) + \frac{f^{''} (1)}{2!} (x - 1)^{2} + \frac{f^{''} (1)}{3!} (x - 1)^{3} + \frac{f^{'''} (1)}{4!} (x - 1)^{4} + \dots

Therefore, first, find the value of the function along with $f^{'} (x), f^{''} (x), f^{''} (x)$ and $f^{''''} (x)$ at $x = 1$

Also, the general of the Taylor series of the function $f$ is

P_{n} (x) = \sum_{k = 0}^{\infty} \frac{f^{k} (x_{0})}{k!} {(x - x_{0})}^{n}

Calculation

So, let's start by constructing the Taylor series table for the function $f (x) = e^{x}$ at $x = 1$

As result, For the function, the Taylor series $f (x) = e^{x}$ $x = 1$ is

$e + e (x - 1) + \frac{e}{2!} (x - 1)^{2} + \frac{e}{3!} (x - 1)^{3} + \frac{e}{4!} (x - 1)^{4} + \dots$

We can also write it as

$P_{n} (x) = \sum_{k = 0}^{\infty} \frac{e}{k!} (x - 1)^{k}$

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Maclaurin and Taylor series: Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series. $e^{x}, x_{0} = 1$

Short Answer

Step by step solution

Given information

Concept

Calculation

Calculation

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Maclaurin and Taylor series: Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.ex,x0=1

Short Answer

Step by step solution

Given information

Concept

Calculation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

Maclaurin and Taylor series: Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series. $e^{x}, x_{0} = 1$