---
title: "Taylor series expansion"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:taylor-series-expansion-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/taylor-series-expansion-two
site: BCT Engineering
---

# Taylor series expansion

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2082 Kartik (new course), 2 marks
- Video: [COMPLEX INTEGRATION | LECTURE 05 | Taylor Series | PRADEEP GIRI SIR](https://www.youtube.com/watch?v=6wo62OVQBzQ) — Pradeep Giri Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/taylor-series-expansion-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Expand f(z) = e^(−z) in a Taylor's series about z = 0.

## Short answer

Taylor series about 0: e^{-z} = ∑_{n=0}^∞ (-1)^n z^n/n! = 1 - z + z^2/2! - …, convergent for all z (radius ∞).

## Answer

Taylor series about z = 0: f(z) = Σ f⁽ⁿ⁾(0) zⁿ/n!. For f(z) = e^(−z), the nth derivative is f⁽ⁿ⁾(z) = (−1)ⁿ e^(−z), so f⁽ⁿ⁾(0) = (−1)ⁿ.

```text
e^(−z) = Σ_{n=0}^∞ (−1)ⁿ zⁿ / n!
       = 1 − z + z²/2! − z³/3! + z⁴/4! − ...
```

**Answer:** e^(−z) = Σ (−z)ⁿ/n!, valid for all z (radius of convergence ∞, since e^(−z) is entire).

## Check yourself

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

1. Taylor series about 0 is:
   A. sum f^(n)(0) z^n
   B. sum f^(n)(0) z^n / n!
   C. sum f(0) z^n/n
   D. sum f^(n)(z) / n!
   Answer: B. Standard Taylor formula. (Key point: Formula)

2. f^(n)(0) for e^(-z) is:
   A. 1
   B. -1
   C. (-1)^n
   D. n!
   Answer: C. Each derivative brings a factor -1. (Key point: Derivative)

3. The series converges:
   A. for |z|<1
   B. for |z|<pi
   C. for Re z>0
   D. for all z, since e^(-z) is entire
   Answer: D. Radius is infinity. (Key point: Convergence)

## Related in Mathematics III

- [Taylor series expansion](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ataylor-series-expansion-three) — Mathematics III, chapter 2
- [Taylor series expansion](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ataylor-series-expansion) — Mathematics III, chapter 2
- [Laurent series expansion](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaurent-series-expansion) — Mathematics III, chapter 2

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