---
title: "Convergence of series (n²−1)/(n²+1)xⁿ"
subject: "Mathematics"
chapter: 6
key: "semester_2/mathematics:convergence-series-n2-1-n2"
source: https://www.bctengineer.com/q/semester_2/mathematics/convergence-series-n2-1-n2
site: BCT Engineering
---

# Convergence of series (n²−1)/(n²+1)xⁿ

Mathematics · Chapter 6: Infinite Series

- Asked in: 2075 Baisakh, 5 marks; 2070 Magh, 5 marks (asked 2 times)
- Video: [D' Alembert's Ratio Test for convergence | complete concept and problem | part 9 |](https://www.youtube.com/watch?v=sPrBk8IwPPs) — Dr Barun Kumar yadav
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/convergence-series-n2-1-n2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-6

## Question

Test the convergence of the series: x + (3/5)x² + (8/10)x³ + (15/17)x⁴ + ... + ((n²−1)/(n²+1))xⁿ + ..., x > 0.

## Short answer

Converges for 0 < x < 1, diverges for x ≥ 1. At x =1, general term tends to1 ≠ 0, so series diverges.

## Answer

[The terms printed are x, (3/5)x², (8/10)x³, (15/17)x⁴, …; coefficients 3/5, 8/10, 15/17 fit (n² − 1)/(n² + 1) for n = 2, 3, 4. The first term printed as x is read as the n = 1 term, which does not change convergence.]

General term: uₙ = (n² − 1)/(n² + 1) · xⁿ.

#### Ratio test
```text
uₙ₊₁/uₙ = x · [((n+1)² − 1)/((n+1)² + 1)] · [(n² + 1)/(n² − 1)]
```
As n → ∞ each bracket tends to 1, so the ratio → x.

- x < 1: **converges**.
- x > 1: **diverges**.

#### The case x = 1
uₙ = (n² − 1)/(n² + 1) = 1 − 2/(n² + 1) → 1 ≠ 0. Since the general term does not tend to zero, the series **diverges**.

**Answer: converges for 0 < x < 1; diverges for x ≥ 1.**

## Check yourself

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

1. Which general term fits the coefficients 3/5, 8/10, 15/17 (n = 2, 3, 4)?
   A. (n² − 1)/(n² + 1)
   B. (n + 1)/(n + 3)
   C. (n² + 1)/(n² − 1)
   D. (n − 1)/(n + 1)
   Answer: A. For n = 2: 3/5, n = 3: 8/10, n = 4: 15/17, so uₙ = (n²−1)/(n²+1) xⁿ. (Key point: General term)

2. What does the ratio test give for x ≠ 1?
   A. Ratio tends to x²
   B. Ratio tends to x: converges for x < 1, diverges for x > 1
   C. Ratio tends to 1: inconclusive for all x
   D. Ratio tends to 1/x
   Answer: B. The coefficient fractions each tend to 1, so uₙ₊₁/uₙ → x. (Key point: Ratio test)

3. Why does the series diverge at x = 1?
   A. It is the harmonic series
   B. The ratio exceeds 1
   C. The general term tends to 1, not 0
   D. It alternates without decreasing
   Answer: C. uₙ = 1 − 2/(n²+1) → 1 ≠ 0, so the series diverges. Hence it converges for 0 < x < 1 and diverges for x ≥ 1. (Key point: Case x = 1)

## Related in Mathematics

- [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 2x+3x²/8+4x³/27+...](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aconvergence-2x-3x2-8-4x3) — 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×
- [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×

## Related in other subjects

- [Balancing reservoir, intermittent supply 7-10 and 17-20](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ares-c74) — Water Supply Engineering, chapter 7
- [Poker test, 10,000 five-digit numbers (2072)](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Apoker-test-numerical-2072-ashwin) — Simulation and Modeling, chapter 6, asked 4×
- [Initial values with switch open to closed (10 Ω, 15 Ω, 2 F)](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Ainitial-conditions-switch-open-10-15) — Electrical Circuit Theory, chapter 2
- [Poker test, 10,000 five-digit numbers (2074)](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Apoker-test-numerical-2074-magh) — Simulation and Modeling, chapter 6, asked 2×