Math Problem Statement

Use the root test to find the radius of convergence for the series \(\sum_{n=1}^\infty \left(\frac{n-1}{2n+4}\right)^n x^n.\)

Solution

To find the radius of convergence of the given series

n=1(n12n+4)nxn,\sum_{n=1}^\infty \left(\frac{n-1}{2n+4}\right)^n x^n,

we use the root test. Here's the solution step-by-step:


Step 1: Root Test Formula

The root test states that for the series

an,\sum a_n,

we compute

L=lim supnann.L = \limsup_{n \to \infty} \sqrt[n]{|a_n|}.

  • If L<1L < 1, the series converges absolutely.
  • If L>1L > 1, the series diverges.
  • If L=1L = 1, the test is inconclusive.

For the series n=1anxn\sum_{n=1}^\infty a_n x^n, the radius of convergence RR is determined by:

1R=lim supnann.\frac{1}{R} = \limsup_{n \to \infty} \sqrt[n]{|a_n|}.


Step 2: Identify ana_n

Here, an=(n12n+4)na_n = \left(\frac{n-1}{2n+4}\right)^n. We focus on an|a_n| since the series includes xnx^n.


Step 3: Apply the Root Test

We compute:

[ \sqrt[n]{|a_n|} = \sqrt[n]{\left(\frac{n-1}{2n+4}\right)^n} = \frac{n-1}{2n+4}. ]

Next, we take the limit superior (lim sup\limsup) as nn \to \infty:

L=limnn12n+4.L = \lim_{n \to \infty} \frac{n-1}{2n+4}.

Simplify the fraction:

n12n+4=n(11n)2n(1+42n)=11n2+4n.\frac{n-1}{2n+4} = \frac{n(1 - \frac{1}{n})}{2n(1 + \frac{4}{2n})} = \frac{1 - \frac{1}{n}}{2 + \frac{4}{n}}.

As nn \to \infty, 1n0-\frac{1}{n} \to 0 and 4n0\frac{4}{n} \to 0, so:

L=12.L = \frac{1}{2}.


Step 4: Account for xnx^n

The series includes xnx^n, so we now include x|x| in the root test. For convergence, we require:

Lx=12x<1.L|x| = \frac{1}{2}|x| < 1.

This simplifies to:

x<2.|x| < 2.

Thus, the radius of convergence is:

R=2.R = 2.


Final Answer:

The radius of convergence is:

2.\boxed{2}.


Would you like more details or have any questions?

Related Questions:

  1. How is the root test different from the ratio test for convergence?
  2. Can you explain why lim sup\limsup is used instead of a standard limit in the root test?
  3. What would change if the series included a factorial term in ana_n?
  4. How would you determine the interval of convergence from the radius of convergence?
  5. How does absolute convergence differ from conditional convergence?

Tip:

Always simplify the fraction inside the lim sup\limsup step carefully, as it determines the result for the radius of convergence.

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Math Problem Analysis

Mathematical Concepts

Series Convergence
Root Test
Radius of Convergence

Formulas

Root Test: L = \limsup_{n \to \infty} \sqrt[n]{|a_n|}
Radius of Convergence: \frac{1}{R} = \limsup_{n \to \infty} \sqrt[n]{|a_n|}

Theorems

Root Test

Suitable Grade Level

Undergraduate Mathematics