Chapter 8: Q. 1 TF (page 671)

Using $f(x) = \sin x$, construct the power series
$\sum_{K = 0}^{\infty} \frac{f^{k} (0)}{k!} x^{k}$

Short Answer

Expert verified

The power series of the given function is :
$x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + . . . . . . . . .$

Step by step solution

Given Information

Given function is $f(x) =\sin x$.

Given power, series is to be constructed

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

Here given form of power series is a Maclaurin series as the point at which the series is to be defined is zero.

$\oint_{a}^{b}x^{3}-x^{2}+\frac{4}{9}x$

Finding Maclaurin series of $f(x)=$sinx

Maclaurin series form is given by:

$\frac{f (0)}{0!} x^{0} + \frac{f^{1} (0)}{1!} x^{1} + \frac{f^{2} (0)}{2!} x^{2} + . . . . . . . .$

Here, f(x) = sin(x)

$f (0) = \sin (0) = 0$

$f^{1} (0) = \cos (0) = 1$

$f^{2} (0) = - \sin (0) = 0$

Substituting the above values in Maclaurin's general equation, we get:

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

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Using \(f(x) = \sin x\), construct the power series
$\sum_{K = 0}^{\infty} \frac{f^{k} (0)}{k!} x^{k}$

Short Answer

Step by step solution

Given Information

Finding Maclaurin series of \(f(x)=\)sinx

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Using \(f(x) = \sin x\), construct the power series∑K=0∞fk(0)k!xk

Short Answer

Step by step solution

Given Information

Finding Maclaurin series of \(f(x)=\)sinx

One App. One Place for Learning.

Most popular questions from this chapter

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Logic and Functions

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Study anywhere. Anytime. Across all devices.

Using \(f(x) = \sin x\), construct the power series
$\sum_{K = 0}^{\infty} \frac{f^{k} (0)}{k!} x^{k}$