---
title: "Convergence of 1+1²/2²+2²/3³+..."
subject: "Mathematics"
chapter: 6
key: "semester_2/mathematics:convergence-1-12-22-22"
source: https://www.bctengineer.com/q/semester_2/mathematics/convergence-1-12-22-22
site: BCT Engineering
---

# Convergence of 1+1²/2²+2²/3³+...

Mathematics · Chapter 6: Infinite Series

- Asked in: 2072 Ashwin, 5 marks; 2068 Magh, 5 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/convergence-1-12-22-22
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-6

## Question

Determine whether the following series is convergent or divergent: 1 + 1²/2² + 2²/3³ + 3³/4⁴ + 4⁴/5⁵ + ...

## Short answer

Diverges. General term uₙ = nⁿ/(n+1)ⁿ⁺¹; limit comparison with 1/n gives 1/e, and the harmonic series diverges.

## Answer

The series is 1 + 1²/2² + 2²/3³ + 3³/4⁴ + 4⁴/5⁵ + …

The pattern of the terms after the first: 1/4, 4/27, 27/256, 256/3125, so the general term is

uₙ = nⁿ/(n + 1)ⁿ⁺¹, n = 1, 2, 3, … (with 1 as the first term, n = 0)

#### Write uₙ in a convenient form
```text
uₙ = (1/(n+1)) · (n/(n+1))ⁿ = (1/(n+1)) · (1 + 1/n)⁻ⁿ
```
Since (1 + 1/n)ⁿ → e, we have (1 + 1/n)⁻ⁿ → 1/e.

#### Limit comparison test with vₙ = 1/n
```text
uₙ/vₙ = (n/(n+1)) (1 + 1/n)⁻ⁿ → 1 · (1/e) = 1/e
```
This limit is finite and non-zero. Σ 1/n is the harmonic series, which diverges, so Σ uₙ also **diverges**.

(The ratio test is inconclusive here: uₙ₊₁/uₙ → 1.)

**Answer: the series diverges.**

## Check yourself

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

1. Which general term fits 1/4, 4/27, 27/256, 256/3125?
   A. nⁿ/(n+1)ⁿ⁺¹
   B. nⁿ/(n+1)ⁿ
   C. n²/(n+1)³
   D. n!/(n+1)ⁿ
   Answer: A. For n = 1: 1/2², n = 2: 4/27 = 2²/3³, so uₙ = nⁿ/(n+1)ⁿ⁺¹. (Key point: General term)

2. Why is the ratio test unsuitable here?
   A. Terms are alternating
   B. uₙ₊₁/uₙ → 1, so it is inconclusive
   C. uₙ₊₁/uₙ → 0, so it fails
   D. The terms are negative
   Answer: B. The ratio tends to 1, which gives no conclusion, so limit comparison is used instead. (Key point: Why not the ratio test)

3. Comparing uₙ with vₙ = 1/n, what is lim uₙ/vₙ and the conclusion?
   A. 1/e; Σ uₙ diverges
   B. e; Σ uₙ converges
   C. 0; Σ uₙ converges
   D. 1/e; Σ uₙ converges
   Answer: A. uₙ/vₙ = (n/(n+1))(1+1/n)⁻ⁿ → 1/e, finite and non-zero, and Σ 1/n diverges, so Σ uₙ diverges. (Key point: Limit comparison)

## Close variants set in other exams

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

## 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×
- [Convergence of 1+x/2+2!x²/3²+...](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aconvergence-1-x-2-2) — Mathematics, chapter 6, asked 2×
- [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×

## Related in other subjects

- [Convergent versus divergent plate boundary](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Aconvergent-versus-divergent-plate-boundary) — Engineering Geology I, chapter 2, asked 3×
- [Convergent-divergent steam nozzle design](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Asteam-nozzle-numerical) — Fluid Machines, chapter 5
- [Convergent-divergent nozzle with efficiency](https://www.bctengineer.com/question.md?key=semester_7%2Fturbo_machines%3Acd-nozzle-efficiency-exit-area-numerical) — Turbo Machines, chapter 4
- [Convergence; coefficients of ±1.5 square wave](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asquare-wave-convergence-coefficients) — Signals and Systems, chapter 2