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Chapter 8: Q. 10 (page 679)

What is the relationship between a Maclaurin series and a power series in x?

Short Answer

Expert verified

$The power series in x is same as the Maclaurin series in x if x_{0} = 0 is considered, that is, a_{k} = \frac{f^{k} (0)}{k!} . The Maclaurin series in x is nothing but a special case of power series, that is, a power series calculated about origin is the Maclaurin series of the function .$

Step by step solution

01

Step 1. Given information is:

A Maclaurin series and a power series in x

02

Step 2. Maclaurin series and Power series

$The power series of the function f around the point x = x_{0} is : f (x) = \sum_{k = 0}^{\infty} \frac{f^{k} (x_{0})}{k!} {(x - x_{0})}^{k} The Maclaurin series of the function f around the point x = 0 is : P (x) = \sum_{k = 0}^{\infty} \frac{f^{k} (0)}{k!} {(x)}^{k}$

03

Step 3. Result

$Therefore, the power series in x is same as the Maclaurin series in x if x_{0} = 0 is considered, that is, a_{k} = \frac{f^{k} (0)}{k!} . The Maclaurin series in x is nothing but a special case of power series, that is, a power series calculated about origin is the Maclaurin series of the function .$

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Most popular questions from this chapter

The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a nonnegative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ . It may be shown that $J_{p} (x)$ is given by the following power series in x:

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

Find and graph the first four terms in the sequence of partial sums of $J_{o} (x)$ .

If $f (x) = 4 x^{3} - 5 x^{2} + 6 x + 1 a n d P_{3} (x)$ is the third Taylor polynomial for f at −1, what is the third remainder $R_{3} (x)$ ? What is $R_{4} (x)$ ? (Hint: You can answer this question without finding any derivatives.)

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_{0}$ . Note: These are the same functions and values as in Exercises 41–48.

$53 . \ln (x), 3$

The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a non-negative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ .It may be shown that $J_{p} (x)$ is given by the following power series in x:

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

What is the interval of convergence for $J_{p} (x)$ where p is a non-negative integer

Find the interval of convergence for power series: $\sum_{k = 1}^{\infty} \frac{{(- 1)}^{k}}{k + 1} x^{k}$

See all solutions

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