Chapter 1: Q11P (page 1)

Question: Show that the Maclaurin series for sin x converges to sin x . Hint: If f (x)= sin $f^{(n + 1)} (X) = \pm \sin x o r \pm \cos x$ , and so $|f^{(n + 1)} (x)| \leq 1$
for all x and all n. Let $n \to \infty$ in (14.2).

Short Answer

Expert verified

Hence prove, Maclaurin Series for sin x converges to sin x .

Step by step solution

Define Maclaurin Series

Maclaurin series is basically a type of power series expansion of the function about the origin, with all the terms having positive values expanded as: $f (x) = f (0) + x f' (0) + \frac{x^{2}}{2} f " (0) + . . . . . . . + \frac{x^{2}}{n} f^{(n)} (0) + . . . . . . . . . .$

Determine the proof for the Macular-in series

The given function is f(x)=sin x

Now, the formula for the remainder is given by:

$R_{n} (x) = \frac{x^{n + 1} f^{(n + 1)} (c)}{(n + 1)}$

Consider the equation $f^{(n + 1)} (x) = \pm \sin x o r \pm \cos x$ with $|f^{(n + 1)} (x)| \leq 1$

So, we taking limit of remainder as follow:

$\lim_{n \to \infty} |R_{n} (x)| = \lim_{n \to \infty} (\begin{matrix} x^{n + 1} \\ (n + 1) \end{matrix})$ = 0

Clearly, the remainder is zero, this implies that the series will converges to itself.

Hence prove, Maclaurin Series for sin x converges to sin x .

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Question: Show that the Maclaurin series for sin x converges to sin x . Hint: If f (x)= sin $f^{(n + 1)} (X) = \pm \sin x o r \pm \cos x$ , and so $|f^{(n + 1)} (x)| \leq 1$
for all x and all n. Let $n \to \infty$ in (14.2).

Short Answer

Step by step solution

Define Maclaurin Series

Determine the proof for the Macular-in series

One App. One Place for Learning.

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Question: Show that the Maclaurin series for sin x converges to sin x . Hint: If f (x)= sinf(n+1)(X)=±sinxor±cosx, and so f(n+1)(x)≤1 for all x and all n. Letn→∞in (14.2).

Short Answer

Step by step solution

Define Maclaurin Series

Determine the proof for the Macular-in series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fluids

Dynamics

Torque and Rotational Motion

Oscillations

Linear Momentum

Famous Physicists

Study anywhere. Anytime. Across all devices.

Question: Show that the Maclaurin series for sin x converges to sin x . Hint: If f (x)= sin $f^{(n + 1)} (X) = \pm \sin x o r \pm \cos x$ , and so $|f^{(n + 1)} (x)| \leq 1$
for all x and all n. Let $n \to \infty$ in (14.2).