---
title: "Interval of convergence Σ2ⁿ(x−3)ⁿ/(n+3)"
subject: "Mathematics"
chapter: 6
key: "semester_2/mathematics:interval-convergence-2-x-3"
source: https://www.bctengineer.com/q/semester_2/mathematics/interval-convergence-2-x-3
site: BCT Engineering
---

# Interval of convergence Σ2ⁿ(x−3)ⁿ/(n+3)

Mathematics · Chapter 6: Infinite Series

- Asked in: 2074 Bhadra, 5 marks; 2073 Magh, 5 marks; 2071 Magh, 5 marks; 2068 Magh, 5 marks (asked 4 times)
- Video: [Power Series & Intervals of Convergence](https://www.youtube.com/watch?v=XHoRBh4hQNU) — Dr. Trefor Bazett
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/interval-convergence-2-x-3
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-6

## Question

Find the interval and radius of convergence of the power series Σ (n=1 to ∞) 2ⁿ(x−3)ⁿ/(n+3).

## Short answer

Radius 1/2; interval of convergence is [5/2, 7/2) (centre 3, left endpoint included, right excluded).

## Answer

The series is Σ uₙ with uₙ = 2ⁿ(x − 3)ⁿ/(n + 3), n = 1, 2, 3, … It is a power series with centre 3.

#### Radius (ratio test)
```text
|uₙ₊₁/uₙ| = 2|x − 3| · (n + 3)/(n + 4) → 2|x − 3| as n → ∞
```
The series converges absolutely if 2|x − 3| < 1, i.e. |x − 3| < 1/2.

**Radius of convergence R = 1/2.**

The open interval is 5/2 < x < 7/2.

#### End points
- x = 7/2: x − 3 = 1/2, so uₙ = 1/(n + 3). Σ 1/(n + 3) behaves like the harmonic series (limit comparison with 1/n), so it **diverges**.
- x = 5/2: x − 3 = −1/2, so uₙ = (−1)ⁿ/(n + 3). This is an alternating series with decreasing terms tending to 0, so it **converges** (Leibniz test).

**Answer: R = 1/2; the interval of convergence is 5/2 ≤ x < 7/2** (centre 3, left end included, right end excluded).

## Check yourself

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

1. For uₙ = 2ⁿ(x−3)ⁿ/(n+3), what does |uₙ₊₁/uₙ| tend to as n → ∞?
   A. |x − 3|
   B. 2|x − 3|
   C. |x − 3|/2
   D. 2
   Answer: B. The factor 2(x−3) stays and (n+3)/(n+4) → 1, so the limit is 2|x − 3|. (Key point: Ratio test limit)

2. Requiring 2|x−3| < 1, what are the radius and open interval?
   A. R = 2, 1 < x < 5
   B. R = 1/2, 5/2 < x < 7/2
   C. R = 1/2, −1/2 < x < 1/2
   D. R = 1, 2 < x < 4
   Answer: B. |x−3| < 1/2 gives R = 1/2, centred at 3, so 5/2 < x < 7/2. (Key point: Radius and open interval)

3. At x = 7/2 the series becomes Σ 1/(n+3). What is its behaviour?
   A. Converges, by the Leibniz test
   B. Converges, as a p-series with p = 1
   C. Diverges, like the harmonic series
   D. Converges absolutely because terms tend to 0
   Answer: C. By limit comparison with 1/n, Σ 1/(n+3) diverges even though its terms tend to 0. (Key point: Endpoint x = 7/2)

4. Which is the interval of convergence?
   A. 5/2 < x < 7/2
   B. 5/2 ≤ x ≤ 7/2
   C. 5/2 < x ≤ 7/2
   D. 5/2 ≤ x < 7/2
   Answer: D. At x = 5/2 the series is alternating and converges (Leibniz), at x = 7/2 it diverges, so the left end is included only. (Key point: Final interval)

## Close variants set in other exams

- [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×
- [Interval, radius and centre of convergence of Σ2ⁿxⁿ/n!](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainterval-radius-centre-convergence-2) — Mathematics, chapter 6, asked 13×

## Related in Mathematics

- [Interval of convergence Σ(−1)ⁿ(x−1)ⁿ/√n](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ainterval-convergence-1-x-1) — Mathematics, chapter 6
- [Convergence of series (n²−1)/(n²+1)xⁿ](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aconvergence-series-n2-1-n2) — Mathematics, chapter 6, asked 2×