Convergence Analysis of a Series with Ratio and Root Tests

Math Problem Statement

Discuss the convergence of series

Solution

To analyze the convergence of the given series:

$\frac{x^2}{2\sqrt{1}} + \frac{x^3}{3\sqrt{2}} + \frac{x^4}{4\sqrt{3}} + \cdots$

We represent the general term of the series and test for convergence.

Step 1: Identify the General Term

The $n$ -th term ( $T_n$ ) of the series is:

$T_n = \frac{x^{n+1}}{(n+1)\sqrt{n}}$

where $n = 1, 2, 3, \dots$ .

Step 2: Test for Convergence

We analyze the series using two tests: the Ratio Test and the Root Test.

(a) Ratio Test:

The ratio test involves computing:

$L = \lim_{n \to \infty} \left| \frac{T_{n+1}}{T_n} \right|$

Substitute $T_n = \frac{x^{n+1}}{(n+1)\sqrt{n}}$ :

$\frac{T_{n+1}}{T_n} = \frac{\frac{x^{n+2}}{(n+2)\sqrt{n+1}}}{\frac{x^{n+1}}{(n+1)\sqrt{n}}}$

Simplify the expression:

= |x| \cdot \frac{\sqrt{n}}{\sqrt{n+1}} \cdot \frac{n+1}{n+2}$$ Take the limit as $$n \to \infty$$: $$L = |x| \cdot \lim_{n \to \infty} \frac{\sqrt{n}}{\sqrt{n+1}} \cdot \frac{n+1}{n+2}$$ - $$\frac{\sqrt{n}}{\sqrt{n+1}} \to 1$$ - $$\frac{n+1}{n+2} \to 1$$ Thus: $$L = |x|$$ - If $$L < 1$$, the series converges absolutely. - If $$L > 1$$, the series diverges. - If $$L = 1$$, the ratio test is inconclusive. #### (b) Root Test: The root test involves computing: $$L = \lim_{n \to \infty} \sqrt[n]{|T_n|}$$ From $$T_n = \frac{x^{n+1}}{(n+1)\sqrt{n}}$$, we have: \[ \sqrt[n]{|T_n|} = \sqrt[n]{|x|^{n+1}} \cdot \sqrt[n]{\frac{1}{(n+1)\sqrt{n}}} \] - $$\sqrt[n]{|x|^{n+1}} = |x| \cdot \sqrt[n]{|x|} \to |x|$$ as $$n \to \infty$$, - $$\sqrt[n]{\frac{1}{(n+1)\sqrt{n}}} \to 1$$ as $$n \to \infty$$. Thus: $$L = |x|$$ - If $$L < 1$$, the series converges absolutely. - If $$L > 1$$, the series diverges. - If $$L = 1$$, the root test is inconclusive. --- ### Step 3: Conclusion Both the ratio and root tests lead to the same criterion. The series converges **absolutely** for: $$|x| < 1$$ and diverges for $$|x| \geq 1$$. Would you like a detailed explanation for a specific step or a different convergence method? --- ### Relative Questions: 1. What is the difference between absolute and conditional convergence? 2. How can the comparison test be applied to this series? 3. What does the convergence depend on if $$x = 1$$ or $$x = -1$$? 4. Could the series be expressed in terms of a standard function (e.g., logarithms or exponentials)? 5. How does the general term's denominator impact convergence? --- ### Tip: When testing convergence of series involving factorials, powers, or roots, the **Ratio Test** is often the most efficient method.

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Math Problem Analysis

Mathematical Concepts

Series Convergence
Ratio Test
Root Test

Formulas

T_n = x^(n+1) / [(n+1) * sqrt(n)]
Ratio Test: L = lim_{n -> ∞} |T_{n+1} / T_n|
Root Test: L = lim_{n -> ∞} nth_root(|T_n|)

Theorems

Ratio Test
Root Test

Suitable Grade Level

Undergraduate Level (Calculus or Advanced Mathematics)

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