---
title: "Interval, radius and centre of convergence of Σ2ⁿxⁿ/n!"
subject: "Mathematics"
chapter: 6
key: "semester_2/mathematics:interval-radius-centre-convergence-2"
source: https://www.bctengineer.com/q/semester_2/mathematics/interval-radius-centre-convergence-2
site: BCT Engineering
---

# Interval, radius and centre of convergence of Σ2ⁿxⁿ/n!

Mathematics · Chapter 6: Infinite Series

- Asked in: 2069 Bhadra, 5 marks
- Video: [How to find the radius and interval of convergence of a power series](https://www.youtube.com/watch?v=iXP8z3pYM_I) — bprp calculus basics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/interval-radius-centre-convergence-2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-6

## Question

Find the interval of convergence, radius of convergence and centre of convergence of the power series Σ 2ⁿxⁿ/n!.

## Short answer

Ratio test: |u_{n+1}/u_n| = 2|x|/(n+1) → 0 for all x. Radius R = ∞, centre at 0, interval (-∞, ∞).

## Answer

The series is Σ (n = 0 to ∞) 2ⁿ xⁿ/n!.

#### Centre
The series is in powers of x = (x − 0), so the **centre of convergence is x = 0**.

#### Radius (ratio test)
```text
|uₙ₊₁/uₙ| = 2|x|/(n + 1) → 0   as n → ∞, for every fixed x
```
The limit 0 < 1 for every x, so the series converges for all x.

**Radius of convergence R = ∞.**

**Interval of convergence: (−∞, ∞).**

(In fact Σ (2x)ⁿ/n! = e^(2x), which is defined for every real x.)

## Check yourself

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

1. What is the centre of convergence of Σ 2ⁿxⁿ/n!?
   A. x = 2
   B. x = 0
   C. x = 1
   D. x = −2
   Answer: B. The series is in powers of x = (x − 0), so the centre is 0. (Key point: Centre)

2. What does |uₙ₊₁/uₙ| = 2|x|/(n+1) tend to as n → ∞?
   A. 2|x|
   B. 1
   C. 0 for every x
   D. ∞
   Answer: C. The factorial makes the denominator grow, so the limit is 0 < 1 for every x. (Key point: Ratio test limit)

3. What are the radius and interval of convergence?
   A. R = 2, (−2, 2)
   B. R = 1/2, (−1/2, 1/2)
   C. R = 0, only x = 0
   D. R = ∞, (−∞, ∞)
   Answer: D. The ratio limit is 0 for all x, so the series converges everywhere; it is e^(2x). (Key point: Radius and interval)

## Close variants set in other exams

- [Interval of convergence Σ2ⁿ(x−3)ⁿ/(n+3)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainterval-convergence-2-x-3) — Mathematics, chapter 6, asked 13×
- [Interval of convergence Σ(−1)ⁿ(x−3)ⁿ/(n+1)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainterval-convergence-1-x-3) — Mathematics, chapter 6, asked 13×
- [Interval of convergence Σ(−1)ⁿxⁿ/(n·2ⁿ)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainterval-convergence-1-x-n) — Mathematics, chapter 6, asked 13×
- [Interval and radius of convergence, 1/(n(n+1))](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainterval-radius-convergence-1-n) — Mathematics, chapter 6, asked 13×
- [Interval of convergence Σ(−3)ⁿxⁿ/√(n+1)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainterval-convergence-3-x-sqrt) — Mathematics, chapter 6, asked 13×

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