---
title: "Convergence of 2x+3x²/8+4x³/27+..."
subject: "Mathematics"
chapter: 6
key: "semester_2/mathematics:convergence-2x-3x2-8-4x3"
source: https://www.bctengineer.com/q/semester_2/mathematics/convergence-2x-3x2-8-4x3
site: BCT Engineering
---

# Convergence of 2x+3x²/8+4x³/27+...

Mathematics · Chapter 6: Infinite Series

- Asked in: 2073 Magh, 5 marks; 2071 Magh, 5 marks; 2068 Bhadra, 5 marks (asked 3 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/convergence-2x-3x2-8-4x3
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-6

## Question

Test the series for convergence or divergence: 2x + 3x²/8 + 4x³/27 + ... + ((n+1)xⁿ/n³) + ... (x > 0).

## Short answer

Converges for 0 < x ≤ 1, diverges for x > 1. Ratio test yields limit x; at x =1, limit comparison with 1/n² shows convergence.

## Answer

The series 2x + 3x²/8 + 4x³/27 + … has general term (the denominators 1, 8, 27 are n³ and the numerators 2, 3, 4 are n + 1):

uₙ = (n + 1)xⁿ/n³, n = 1, 2, 3, …

#### Ratio test
```text
uₙ₊₁/uₙ = [(n+2)xⁿ⁺¹/(n+1)³] · [n³/((n+1)xⁿ)]
        = x · (n+2) n³ / (n+1)⁴
        → x as n → ∞
```
- x < 1: converges.
- x > 1: diverges.

#### The case x = 1
uₙ = (n + 1)/n³. Compare with vₙ = 1/n²: uₙ/vₙ = (n + 1)/n → 1, finite and non-zero. Σ 1/n² converges (p-series, p = 2 > 1), so by the limit comparison test Σ uₙ **converges**.

**Answer: the series converges for 0 < x ≤ 1 and diverges for x > 1.**

## Check yourself

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

1. In 2x + 3x²/8 + 4x³/27 + ..., what is the general term?
   A. (n+1)xⁿ/n²
   B. (n+1)xⁿ/n³
   C. (n+1)xⁿ/2ⁿ
   D. n xⁿ/(n+1)³
   Answer: B. Denominators 1, 8, 27 are n³ and numerators 2, 3, 4 are n+1. (Key point: Reading the general term)

2. The ratio uₙ₊₁/uₙ = x(n+2)n³/(n+1)⁴ tends to what, and what follows?
   A. x; convergent for x < 1, divergent for x > 1
   B. 1; always convergent
   C. x; convergent for x > 1
   D. 0; convergent for all x
   Answer: A. The rational factor tends to 1, so the limit is x. The series converges for x < 1 and diverges for x > 1. (Key point: Ratio test conclusion)

3. At x = 1, uₙ = (n+1)/n³. Which comparison settles it?
   A. Compare with 1/n; diverges
   B. Compare with 1/n²; the ratio tends to 1, so it converges
   C. Compare with 1/n³; diverges
   D. Alternating series test
   Answer: B. uₙ/(1/n²) = (n+1)/n → 1 and Σ 1/n² is a convergent p-series, so the series converges for 0 < x ≤ 1. (Key point: Case x = 1)

## Related in Mathematics

- [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×
- [Power series: y''−4xy'+(4x²−2)y=0](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apower-series-y-4xy-4x2) — Mathematics, chapter 4, asked 12×
- [Convergence of x/1.2 + x²/2.3 + x³/3.4](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aconvergence-x-1-2-x2) — Mathematics, chapter 6, asked 3×
- [Convergence of Σ(n+1)ⁿxⁿ/n^(n+1)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aconvergence-n-1-x-n) — Mathematics, chapter 6
- [Convergence of 1/2+(2/3)x+(3/4)²x²+(4/5)³x³+...](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aconvergence-1-2-2-3) — Mathematics, chapter 6
- [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×

## Related in other subjects

- [Big-M method min:Z=4x₁+2x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Abig-m-method-min-z-4x) — Mathematics III, chapter 5
- [Graphical method min:z=4x₁+3x₂+x₃](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Agraphical-method-min-z-4x-3x) — Mathematics III, chapter 5
- [Big-M method max:P=2x₁+x₂+3x₃](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Abig-m-method-max-p-2x) — Mathematics III, chapter 5, asked 17×
- [Duality LPP min:Z=3x₁+2x₂](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aduality-lpp-min-z-3x-2x) — Mathematics III, chapter 5, asked 3×