---
title: "First four terms of Taylor series about z = 2"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:taylor-series-z-plus-1-about-2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/taylor-series-z-plus-1-about-2
site: BCT Engineering
---

# First four terms of Taylor series about z = 2

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2071 Magh, 5 marks
- Video: [f(z)=z+1/(z-3)(z-4) at z=2 by using Taylor's Theorem](https://www.youtube.com/watch?v=yOZN-uxpdG4) — MB Uday
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/taylor-series-z-plus-1-about-2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Find the first four terms of the Taylor's series expansion of the complex function f(z) = (z + 1)/((z - 3)(z - 4)) about the centre z = 2.

## Answer

**Taylor series:** f(z) = Σ aₙ (z − a)ⁿ about the centre z = a = 2, valid for |z − 2| < R, R = distance to the nearest singularity.

f(z) = (z + 1)/((z − 3)(z − 4)) has poles at 3 and 4. The nearest is z = 3, at distance 1, so R = 1.

#### Partial fractions
(z + 1)/((z − 3)(z − 4)) = A/(z − 3) + B/(z − 4)

A = (3 + 1)/(3 − 4) = −4,  B = (4 + 1)/(4 − 3) = 5.

f(z) = −4/(z − 3) + 5/(z − 4).

#### Expand about z = 2
Let t = z − 2. Then z − 3 = t − 1 and z − 4 = t − 2.

```text
−4/(t − 1) = 4/(1 − t)          = 4 Σ tⁿ                (|t| < 1)
 5/(t − 2) = −(5/2)/(1 − t/2)   = −(5/2) Σ (t/2)ⁿ       (|t| < 2)
```

f(z) = Σ_{n=0}^{∞} [ 4 − 5/2ⁿ⁺¹ ] (z − 2)ⁿ

#### First four terms
```text
n = 0 : 4 − 5/2  = 3/2
n = 1 : 4 − 5/4  = 11/4
n = 2 : 4 − 5/8  = 27/8
n = 3 : 4 − 5/16 = 59/16
```

**f(z) = 3/2 + (11/4)(z − 2) + (27/8)(z − 2)² + (59/16)(z − 2)³ + …,  |z − 2| < 1.**

Check: f(2) = 3/((−1)(−2)) = 3/2 ✓.

## Check yourself

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

1. Why is the series about z = 2 valid for |z-2| < 1?
   A. The farthest pole z = 4 is at distance 2, halved
   B. The centre z = 2 is itself a pole
   C. Both poles are at distance 1
   D. The nearest singularity z = 3 is at distance 1 from the centre
   Answer: D. R equals the distance to the nearest singularity; poles are 3 and 4, so R = 1. (Key point: Radius of convergence)

2. What are the partial fractions of (z+1)/((z-3)(z-4))?
   A. -4/(z-3) + 5/(z-4)
   B. 4/(z-3) + 5/(z-4)
   C. -4/(z-3) - 5/(z-4)
   D. 5/(z-3) - 4/(z-4)
   Answer: A. A = (3+1)/(3-4) = -4 and B = (4+1)/(4-3) = 5. (Key point: Partial fractions)

3. With t = z-2, how is 5/(z-4) expanded?
   A. 5 Σ t^n
   B. -(5/2)·1/(1 - t/2) = -(5/2) Σ (t/2)^n
   C. (5/2) Σ (t/2)^n
   D. -5 Σ (t/2)^n
   Answer: B. z-4 = t-2 = -2(1 - t/2), giving -(5/2) Σ (t/2)^n. (Key point: Expansion in t)

4. What is the general coefficient a_n and the first term?
   A. a_n = 4 + 5/2^{n+1}; a_0 = 13/2
   B. a_n = 4 - 5/2^n; a_0 = -1
   C. a_n = 4 - 5/2^{n+1}; a_0 = 3/2
   D. a_n = 5/2^{n+1} - 4; a_0 = -3/2
   Answer: C. Adding 4Σt^n and -(5/2)Σ(t/2)^n gives 4 - 5/2^{n+1}; for n=0 this is 3/2 = f(2). (Key point: Coefficients)

## Related in Applied Mathematics

- [State and prove Taylor's series](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ataylor-series-prove) — Applied Mathematics, chapter 1, asked 2×
- [Taylor series of 1/(z²+4) about z = i](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ataylor-series-1-over-z2-plus-4-about-i) — Applied Mathematics, chapter 1, asked 5×
- [Taylor series of (2z³+1)/(z²+z) about z = 1](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ataylor-series-2z3-about-1) — Applied Mathematics, chapter 1, asked 5×
- [Taylor series of 1/(1-z) about z = 3i](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ataylor-series-1-over-1-z-about-3i) — Applied Mathematics, chapter 1
- [Complex Fourier series; verify convolution](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-series-complex-convolution) — Applied Mathematics, chapter 4
- [Laurent series of 1/(z²-3z+2), 1<|z|<2](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaurent-1-over-z2-3z-2) — Applied Mathematics, chapter 1, asked 2×

## Related in other subjects

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