Q. 58 Find the Maclaurin series for th... [FREE SOLUTION]

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Chapter 8: Q. 58 (page 693)

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.
$\frac{e^{x} - e^{- x}}{2}$

Short Answer

Expert verified

The answer is $\frac{e^{x} - e^{- x}}{2} = {鈭�}_{k = 0}^{鈭�} \frac{1}{(2 k + 1)!} x^{2 k + 1}$

Step by step solution

Step 1. Given Information

Consider the function $\frac{e^{x} - e^{- x}}{2}$

Find the interval of convergence for the series.

We know that the Maclaurin series for the func $e^{x} = {鈭�}_{k = 0}^{鈭�} \frac{1}{k!} x^{k}$

So, the series for $h (x) = e^{- x}$ can be found by substituting $x$ by $- x$

That is, $e^{- x} = {鈭�}_{k = 0}^{鈭�} \frac{1}{k!} {(- x)}^{k}$

Finally, to find the series for the function $f (x) = \frac{e^{x} - e^{- x}}{2}$ we substract the series for $e^{- x}$ from $e^{x}$ and divide the result by 2

Thus, role="math" localid="1649870849460" $\begin{array}{rcl} e^{x} - e^{- x} & = & {鈭�}_{k = 0}^{鈭�} \frac{1}{k!} x^{k} - {鈭�}_{k = 0}^{鈭�} \frac{1}{k!} {(- x)}^{k} \\ = & {鈭�}_{k = 0}^{鈭�} \frac{1}{k!} [x^{k} - (- x^{k})] \\ = & 2 x + \frac{2 x^{3}}{3!} + \frac{2 x^{5}}{5!} + \frac{2 x^{7}}{7!} + . . . \end{array}$

Then dividing by $2$ we get

$\frac{e^{x} - e^{- x}}{2} = x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \frac{x^{7}}{7!} + . . .$

implies that,role="math" localid="1649871145815" $\frac{e^{x} - e^{- x}}{2} = {鈭�}_{k = 0}^{鈭�} \frac{1}{(2 k + 1)!} x^{2 k + 1}$

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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.
$\frac{e^{x} - e^{- x}}{2}$

Short Answer

Step by step solution

Step 1. Given Information

Find the interval of convergence for the series.

One App. One Place for Learning.

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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.ex-e-x2

Short Answer

Step by step solution

Step 1. Given Information

Find the interval of convergence for the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Pure Maths

Statistics

Logic and Functions

Mechanics 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.
$\frac{e^{x} - e^{- x}}{2}$