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

# Convergence of Σ n/(1+n√(n+1))

Mathematics · Chapter 6: Infinite Series

- Asked in: 2072 Magh, 5 marks
- Video: [Infinite Series - Comparison Test For Convergence of Series | Calculus](https://www.youtube.com/watch?v=S-qzRWJOlf8) — Dr.Gajendra Purohit
- 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-nsqrt-n
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-6

## Question

Determine whether the series Σ n/(1+n√(n+1)) is convergent or divergent.

## Short answer

Term behaves like 1/√n. Limit comparison with ∑1/√n gives limit 1; the p-series ∑1/√n diverges (p=1/2), so the series diverges.

## Answer

The series is Σ uₙ with uₙ = n/(1 + n√(n + 1)).

#### Behaviour for large n
In the denominator, n√(n + 1) ≈ n^(3/2), so uₙ ≈ n/n^(3/2) = 1/√n.

#### Limit comparison test with vₙ = 1/√n
```text
uₙ/vₙ = n√n / (1 + n√(n+1))
      = 1 / (1/n^(3/2) + √(1 + 1/n))
      → 1 / (0 + 1) = 1
```
The limit is finite and positive, so Σ uₙ and Σ vₙ behave alike. Σ 1/√n = Σ 1/n^(1/2) is a p-series with p = 1/2 ≤ 1, which diverges.

**Answer: the series Σ n/(1 + n√(n + 1)) diverges.**

## Check yourself

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

1. For large n, uₙ = n/(1 + n√(n+1)) behaves like?
   A. 1/n
   B. 1/√n
   C. 1/n²
   D. 1
   Answer: B. The denominator is about n^(3/2), so uₙ ≈ n/n^(3/2) = 1/√n. (Key point: Dominant behaviour)

2. With vₙ = 1/√n, what is lim uₙ/vₙ?
   A. 0
   B. ∞
   C. 1
   D. 1/2
   Answer: C. uₙ/vₙ = 1/(1/n^(3/2) + √(1+1/n)) → 1, finite and positive. (Key point: Limit comparison)

3. Why does Σ uₙ diverge?
   A. Σ 1/√n is a p-series with p = 1/2 ≤ 1, which diverges
   B. Terms do not tend to 0
   C. Ratio test gives limit greater than 1
   D. Σ 1/√n converges, so uₙ is bounded
   Answer: A. Limit comparison makes Σ uₙ behave like the divergent p-series with p = 1/2. (Key point: Conclusion)

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