Chapter 8: Q.8 (page 659)

If $f$ is a function such that $f (0) = 1$ and $f^{'} (x) = f (x)$ for every value of $x$ , find the Maclaurin series for $f$ .

Short Answer

Expert verified

The Maclaurin series for the function is:

$f (x) = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots$

Or, it can be written as

$e^{x} = \sum_{k = 0}^{\infty} \frac{1}{k!} x^{k}$

Step by step solution

Given Information

Given equations :

$f (0) = 1 f' (x) = f (x)$

Finding the Maclaurin series for f

Let us consider the function such that $f (0) = 1 and f' (x) = f (x)$ and

So, the function must be $f (x) = e^{x}$

Since, the general formula to calculate the Maclaurin series for the function is:

$f (x) = f (0) + f^{'} (0) x + \frac{f^{''} (0)}{2!} x^{2} + \frac{f^{''} (0)}{3!} x^{3} + \dots$

As role="math" localid="1650439558986" $f (0) = 1$ , therefore role="math" localid="1650439565403" $f^{'} (0), f^{''} (0) and f'' (0) are 1$

So, The Maclaurin series for the function is:

role="math" localid="1650439570178" $f (x) = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots \Rightarrow e^{x} = \sum_{k = 0}^{\infty} \frac{1}{k!} x^{k}$

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 Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

If $f$ is a function such that $f (0) = 1$ and $f^{'} (x) = f (x)$ for every value of $x$ , find the Maclaurin series for $f$ .

Short Answer

Step by step solution

Given Information

Finding the Maclaurin series for f

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

If fis a function such that f(0)=1and f'(x)=f(x)for every value of x, find the Maclaurin series for f.

Short Answer

Step by step solution

Given Information

Finding the Maclaurin series for f

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

If $f$ is a function such that $f (0) = 1$ and $f^{'} (x) = f (x)$ for every value of $x$ , find the Maclaurin series for $f$ .