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

# Taylor series expansion

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

- Asked in: 2082 Chaitra (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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Find the Taylor series expansion of f(z) = 1/(z − 3) centered at the point z = 4i.

## Short answer

Let w=z-4i. 1/(z-3)=1/(w+(4i-3)) = -∑_{n=0}^∞ (z-4i)^n/(3-4i)^(n+1), valid for |z-4i|<5 (distance to the pole at z=3).

## Answer

Let w = z − 4i. Then z − 3 = w + (4i − 3). Put a = 4i − 3, so |a| = 5.

```text
f(z) = 1/(z − 3) = 1/(a + w) = (1/a) · 1/(1 + w/a)
     = (1/a) Σ_{n=0}^∞ (−w/a)ⁿ         for |w| < |a| = 5
     = Σ (−1)ⁿ (z − 4i)ⁿ / (4i − 3)^(n+1)
```

Since (4i − 3)^(n+1) = (−1)^(n+1) (3 − 4i)^(n+1), this simplifies to

```text
f(z) = − Σ_{n=0}^∞ (z − 4i)ⁿ / (3 − 4i)^(n+1)
     = −(3 + 4i)/25 − (3 + 4i)²/625 · (z − 4i) − ...
```

**Answer:** f(z) = − Σ (z − 4i)ⁿ/(3 − 4i)^(n+1), valid for |z − 4i| < 5 (the distance from 4i to the singularity z = 3).

## Check yourself

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

1. Taylor series about 4i uses w =
   A. z + 4i
   B. z - 3
   C. z
   D. z - 4i
   Answer: D. Expansion in powers of z - 4i. (Key point: Substitution)

2. 1/(z-3) is rewritten as:
   A. (1/a) * 1/(1 + w/a), a = 4i - 3
   B. 1/(1 - w)
   C. (1/w) * 1/(1+a)
   D. 1/a * 1/(1 - a/w)
   Answer: A. z - 3 = a + w. (Key point: Rewrite)

3. The series converges for:
   A. |z - 4i| < 3
   B. |z - 4i| < 5
   C. |z| < 5
   D. all z
   Answer: B. Distance from 4i to the singularity 3 is 5. (Key point: Radius)

## Related in Mathematics III

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