---
title: "Maclaurin expansion of e^x log(1+x)"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:maclaurin-ex-log-1-plus-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/maclaurin-ex-log-1-plus-x
site: BCT Engineering
---

# Maclaurin expansion of e^x log(1+x)

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2073 Chaitra
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/maclaurin-ex-log-1-plus-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Expand eˣ log_e(1+x) in ascending powers of x up to the term containing x⁴ in Maclaurin's series.

## Short answer

Multiply the Maclaurin series of e^x and log(1+x): e^x = 1 + x + x^2/2 + x^3/6 + x^4/24 + ... and log(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ... Collecting terms up to x^4 gives e^x log(1+x) = x + x^2/2 + x^3/3 + 0·x^4 + ... The x^4 term vanishes.

## Answer

Maclaurin's series: f(x) = f(0) + x f′(0) + (x²/2!) f″(0) + ... For products of standard functions we multiply their Maclaurin series (valid by uniqueness of the expansion).

```text
eˣ        = 1 + x + x²/2 + x³/6 + x⁴/24 + ...
log(1+x)  = x − x²/2 + x³/3 − x⁴/4 + ...
```
Multiply and collect terms up to x⁴:

```text
x¹ :  1                                   = 1
x² :  −1/2 + 1                            = 1/2
x³ :  1/3 − 1/2 + 1/2                     = 1/3
x⁴ :  −1/4 + 1/3 − 1/4 + 1/6              = 0
```
(x³ term: from 1·(x³/3), x·(−x²/2) and (x²/2)·x. x⁴ term: from 1·(−x⁴/4), x·(x³/3), (x²/2)(−x²/2) and (x³/6)·x.)

**Answer: eˣ log(1+x) = x + x²/2 + x³/3 + 0·x⁴ + ...**, so the x⁴ coefficient is zero.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is e^x log(1+x) expanded?
   A. Multiply the Maclaurin series of e^x and log(1+x)
   B. Add the two series
   C. Differentiate four times only
   D. Use L'Hospital
   Answer: A. Uniqueness of the expansion allows series multiplication. (Key point: Method)

2. The x^2 coefficient is:
   A. 1
   B. 3/2
   C. 1/2
   D. -1/2
   Answer: C. -1/2 + 1. (Key point: Coefficient of x^2)

3. The x^3 coefficient is:
   A. 1/3
   B. 1/2
   C. 0
   D. 1/6
   Answer: A. 1/3 - 1/2 + 1/2. (Key point: Coefficient of x^3)

4. The x^4 coefficient is:
   A. 0
   B. -1/4
   C. 1/12
   D. 1/4
   Answer: A. -1/4 + 1/3 - 1/4 + 1/6 = 0. (Key point: Coefficient of x^4)

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