---
title: "Convergence of Σ(n+1)ⁿxⁿ/n^(n+1)"
subject: "Mathematics"
chapter: 6
key: "semester_2/mathematics:convergence-n-1-x-n"
source: https://www.bctengineer.com/q/semester_2/mathematics/convergence-n-1-x-n
site: BCT Engineering
---

# Convergence of Σ(n+1)ⁿxⁿ/n^(n+1)

Mathematics · Chapter 6: Infinite Series

- Asked in: 2069 Poush, 5 marks
- Video: [Choosing Which Convergence Test to Apply to 8 Series](https://www.youtube.com/watch?v=0wefqjpQyKM) — 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/convergence-n-1-x-n
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-6

## Question

Test the convergence of the series Σ (n+1)ⁿ xⁿ / n^(n+1).

## Short answer

Ratio test: limit |u_{n+1}/u_n| = x; converges for 0 < x < 1. At x=1, term ~ e/n, diverges by comparison with harmonic series. Diverges for x ≥ 1.

## Answer

[Read as x > 0, as in similar questions; the general term is uₙ = (n + 1)ⁿ xⁿ / n^(n+1).]

#### Simplify
uₙ = (n + 1)ⁿ xⁿ/n^(n+1) = (1/n) · ((n + 1)/n)ⁿ · xⁿ = (1/n)(1 + 1/n)ⁿ xⁿ.

#### Ratio test
```text
uₙ₊₁/uₙ = x · [n/(n+1)] · [(1 + 1/(n+1))ⁿ⁺¹ / (1 + 1/n)ⁿ] → x · 1 · (e/e) = x
```
- x < 1: **converges**.
- x > 1: **diverges**.

#### The case x = 1
uₙ = (1/n)(1 + 1/n)ⁿ. Compare with vₙ = 1/n: uₙ/vₙ = (1 + 1/n)ⁿ → e, finite and positive. Σ 1/n diverges, so Σ uₙ **diverges**.

**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. Which form is uₙ = (n+1)ⁿxⁿ/n^(n+1) simplified to?
   A. (1/n)(1 + 1/n)ⁿ xⁿ
   B. (1/n)(1 − 1/n)ⁿ xⁿ
   C. n(1 + 1/n)ⁿ xⁿ
   D. (1 + 1/n)^(n+1) xⁿ
   Answer: A. (n+1)ⁿ/n^(n+1) = (1/n)((n+1)/n)ⁿ = (1/n)(1+1/n)ⁿ. (Key point: Simplified general term)

2. What does uₙ₊₁/uₙ tend to?
   A. x/e
   B. x
   C. ex
   D. 1
   Answer: B. The factors give n/(n+1) → 1 and (1+1/(n+1))^(n+1)/(1+1/n)ⁿ → e/e = 1, so the limit is x. (Key point: Ratio test limit)

3. At x = 1, uₙ = (1/n)(1 + 1/n)ⁿ. What is the result?
   A. Converges, compare with 1/n²
   B. Diverges, since uₙ/(1/n) → e and Σ 1/n diverges
   C. Converges, since (1+1/n)ⁿ is bounded
   D. Diverges, since terms do not tend to 0
   Answer: B. Limit comparison with 1/n gives e, finite and positive, so it diverges. Hence converges for 0 < x < 1 only. (Key point: Case x = 1)

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