How would I find the Maclaurian series for f(x)? (attached) **Problem Statement:**Find a Maclaurin series for $ f(x) $. (Use $ \frac{(2n)!}{2^n n! (2n-1)} $ for $ 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-3) $.)**Function Definition:**\[ f(x) = \int_0^x \sqrt{1 + t^3} \, dt \]**Maclaurin Series Expansion:**\[ f(x) = x + \frac{x^4}{8} + \sum_{n=2}^{\infty} \text{[Expression Missing]} \]**Explanation:**The problem involves finding the Maclaurin series of a function $ f(x) $ defined as the integral from 0 to $ x $ of $ \sqrt{1 + t^3} $. The series expansion starts with terms $ x $ and $ \frac{x^4}{8} $, followed by an infinite series starting from $ n=2 $. The exact expression for the general term in the series is not provided in the image. Note: Understanding the pattern and determining the full series calculation requires further solving steps not present in the image.

given fx=∫0x1+t3dt to find the maclaurin series

Answered: Find a Maclaurin series for f(x). (Use…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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How would I find the Maclaurian series for f(x)? (attached)

**Problem Statement:**

Find a Maclaurin series for $ f(x) $. (Use $ \frac{(2n)!}{2^n n! (2n-1)} $ for $ 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-3) $.)

**Function Definition:**

\[ f(x) = \int_0^x \sqrt{1 + t^3} \, dt \]

**Maclaurin Series Expansion:**

\[ f(x) = x + \frac{x^4}{8} + \sum_{n=2}^{\infty} \text{[Expression Missing]} \]

**Explanation:**

The problem involves finding the Maclaurin series of a function $ f(x) $ defined as the integral from 0 to $ x $ of $ \sqrt{1 + t^3} $. The series expansion starts with terms $ x $ and $ \frac{x^4}{8} $, followed by an infinite series starting from $ n=2 $. The exact expression for the general term in the series is not provided in the image.

Note: Understanding the pattern and determining the full series calculation requires further solving steps not present in the image.

Expert Solution

Step 1

given

$f (x) = \int_{0}^{x} \sqrt{1 + t^{3}} d t$

to find the maclaurin series

Step 2

$\sqrt{1 + t^{3}} = 1 + \sum_{k = 1}^{\infty} \frac{\frac{1}{2} (\frac{1}{2} - 1) . . . . . . (\frac{1}{2} - k + 1)}{k!} t^{3 k} = 1 + \sum_{k = 1}^{\infty} \frac{1 (- 1) (- 3) . . . . . . (- 2 k + 3)}{2^{k} k!} t^{3 k} = 1 + \sum_{k = 1}^{\infty} \frac{{(- 1)}^{k} (2 k)!}{2^{k} k! (2 k - 1) k!} t^{3 k} = 1 + \sum_{k = 1}^{\infty} \frac{{(- 1)}^{k + 1} (2 k)!}{2^{2 k} k! (2 k - 1) k!} t^{3 k} = 1 + \sum_{k = 1}^{\infty} \frac{{(- 1)}^{k + 1} (2 k)!}{4^{k} {(k!)}^{2} (2 k - 1)} t^{3 k}$

substituting the value of $\sqrt{1 + t^{3}}$ into $f (x) = \int_{0}^{x} \sqrt{1 + t^{3}} d t$

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