### Problem Statement(a) **Find the Taylor series for** $ f(x) = \ln(1 + 4x) $ **about** $ x = 0 $ **by taking derivatives.**\[\sum_{n=1}^{\infty} \quad \text{(Input your series here)}\](b) **Compare your result above to the series for** $ \ln(1 + x) $. **How could you have obtained your answer to part (a) from the series for** $ \ln(1 + x) $?(c) **What do you expect the interval of convergence for the series for** $ \ln(1 + 4x) $ **to be?**\[\underline{\quad} < x \leq \underline{\quad}\]### ExplanationThis exercise involves finding the Taylor series for the natural logarithm function centered at $ x = 0 $. You are asked to:1. Compute the Taylor series for $ \ln(1 + 4x) $ by differentiating and finding coefficients.2. Compare it with the known Taylor series for $ \ln(1 + x) $.3. Provide the interval of convergence for the series you found.This exercise assesses your understanding of series expansion using differentiation and your ability to identify patterns and convergence issues in series.

Find the Taylor series for f(x) = ln(1+4x) about x = 0 by taking derivatives. Σ n=1 Compare your result above to the series for ln(1 +æ). How could you have obtained your answer to part (a) from the series for In(1+x)? What do you expect the interval of convergence for the series for In(1+ 4x) to be? < x <

Find the Taylor series for f(x) = ln(1+4x) about x = 0 by taking derivatives. Σ n=1 Compare your result above to the series for ln(1 +æ). How could you have obtained your answer to part (a) from the series for In(1+x)? What do you expect the interval of convergence for the series for In(1+ 4x) to be? < x <

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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Question

### Problem Statement

(a) **Find the Taylor series for** $ f(x) = \ln(1 + 4x) $ **about** $ x = 0 $ **by taking derivatives.**

\[
\sum_{n=1}^{\infty} \quad \text{(Input your series here)}
\]

(b) **Compare your result above to the series for** $ \ln(1 + x) $. **How could you have obtained your answer to part (a) from the series for** $ \ln(1 + x) $?

(c) **What do you expect the interval of convergence for the series for** $ \ln(1 + 4x) $ **to be?**

\[
\underline{\quad} < x \leq \underline{\quad}
\]

### Explanation

This exercise involves finding the Taylor series for the natural logarithm function centered at $ x = 0 $. You are asked to:

1. Compute the Taylor series for $ \ln(1 + 4x) $ by differentiating and finding coefficients.
2. Compare it with the known Taylor series for $ \ln(1 + x) $.
3. Provide the interval of convergence for the series you found.

This exercise assesses your understanding of series expansion using differentiation and your ability to identify patterns and convergence issues in series.

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