Q. 53 Find the Maclaurin series for th... [FREE SOLUTION]

Chapter 8: Q. 53 (page 659)

Find the Maclaurin series for the functions in Exercises 51鈥�60
by substituting into a known Maclaurin series. Also, give the
interval of convergence for the series.
$\sin (- 5 x^{2})$

Short Answer

Expert verified

The interval of convergence for the series $\begin{array}{rcl} {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k + 2 k + 1} 5^{2 k + 1}}{(2 k + 1)!} x^{4 k + 2} \end{array}$ .

Step by step solution

Step 1. Given Information

The function $f (x) = \sin (- 5 x^{2})$

The objective is to find the series of intervals of convergence .

Step 2. Calculation

So, to get the Maclaurin series for the function $f (x) = \sin (- 5 x^{2})$ , we replace $x$ by $- 5 x^{2}$ in the Maclaurin series of the function $\sin x$

localid="1649949657371" role="math" $\begin{array}{rcl} \sin (- 5 x^{2}) & = & {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k}}{(2 k + 1)!} {(- 5 x^{2})}^{2 k + 1} \\ = & {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k} {(- 1)}^{2 k + 1} 5^{2 k + 1}}{(2 k + 1)!} x^{4 k + 2} \\ = & {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k + 2 k + 1} 5^{2 k + 1}}{(2 k + 1)!} x^{4 k + 2} \end{array}$

Therefore, the required series is $\begin{array}{rcl} {鈭�}_{k = 0}^{鈭�} \frac{{(- 1)}^{k + 2 k + 1} 5^{2 k + 1}}{(2 k + 1)!} x^{4 k + 2} \end{array}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the Maclaurin series for the functions in Exercises 51鈥�60
by substituting into a known Maclaurin series. Also, give the
interval of convergence for the series.
$\sin (- 5 x^{2})$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Applied Mathematics

Calculus

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Find the Maclaurin series for the functions in Exercises 51鈥�60by substituting into a known Maclaurin series. Also, give theinterval of convergence for the series.sin(-5x2)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Applied Mathematics

Calculus

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series for the functions in Exercises 51鈥�60
by substituting into a known Maclaurin series. Also, give the
interval of convergence for the series.
$\sin (- 5 x^{2})$