Chapter 1: Q13P (page 1)

Question: Show that the Maclaurin series for ${(1 + x)}^{p}$ converges to ${(1 + x)}^{p}$ when $0 < x < 1$

Short Answer

Expert verified

Hence prove, maclaurin Series for converges to

Step by step solution

Maclaurin Series

Maclaurin series is basically a type of power series expansion of the function f(x) about the origin, with all the terms having positive values expanded as:

Determine the proof

The given function is

Now, the general formula for the remainder of any function is given by:

Let us find the derivative as follow:

Solve further as:

\Rightarrow f^{n + 1} (x) = \frac{p {(1 + x)}^{- n}}{(p - n + 1)} \{a = 0\}

Determine the proof

Let,

Then, the function becomes:

Now, the remainder will be transformed to:

So, taking limit of remainder as follow:

$\lim_{n \to \infty} |R_{n} (x)| = \lim_{n \to \infty} [\begin{matrix} {(x)}^{n + 1} 2^{n} . p \\ 3^{n} . (n + 1) . (p - n + 1) \end{matrix}] = 0$

\{\lim_{n \to \infty} {(x)}^{n + 1} = 0, \lor 0 < x < 1\}

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

Hence prove, Maclaurin Series for converges to.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Question: Show that the Maclaurin series for ${(1 + x)}^{p}$ converges to ${(1 + x)}^{p}$ when $0 < x < 1$

Short Answer

Step by step solution

Maclaurin Series

Determine the proof

Determine the proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Kinematics Physics

Dynamics

Waves Physics

Force

Magnetism and Electromagnetic Induction

Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.

Question: Show that the Maclaurin series for (1+x)pconverges to(1+x)pwhen0<x<1

Short Answer

Step by step solution

Maclaurin Series

Determine the proof

Determine the proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Kinematics Physics

Dynamics

Waves Physics

Force

Magnetism and Electromagnetic Induction

Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.

Question: Show that the Maclaurin series for ${(1 + x)}^{p}$ converges to ${(1 + x)}^{p}$ when $0 < x < 1$