Math Problem Statement

Find the radius of convergence and interval of convergence of the given series using the ratio test. \[ \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \]

Solution

The given series is:

∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}

To find the radius of convergence RR and the interval of convergence, we can use the Ratio Test. The Ratio Test states that for a series ∑an\sum a_n, the radius of convergence RR can be found by examining the limit:

lim⁡n→∞∣an+1an∣\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|

Step 1: Identify ana_n

In this series, we have: an=(−1)nx2n(2n)!a_n = \frac{(-1)^n x^{2n}}{(2n)!}

Step 2: Apply the Ratio Test

Using the Ratio Test, we calculate:

∣an+1an∣=∣(−1)n+1x2(n+1)(2(n+1))!(−1)nx2n(2n)!∣\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{(-1)^{n+1} x^{2(n+1)}}{(2(n+1))!}}{\frac{(-1)^n x^{2n}}{(2n)!}} \right|

Simplify this expression:

= \left| \frac{x^2}{(2n+2)(2n+1)} \right|$$ As $$n \to \infty$$, the terms $$(2n+2)(2n+1)$$ grow large, causing the fraction to approach zero: $$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 0$$ ### Step 3: Interpret the Result Since this limit is zero (which is less than 1) for all values of $$x$$, the series converges for all $$x$$ values. ### Conclusion The radius of convergence $$R$$ is infinite, meaning the series converges for all real $$x$$. Therefore, the **interval of convergence** is: $$(-\infty, \infty)$$ Would you like a further breakdown or have any specific questions? --- **Related Questions:** 1. What would happen to the radius of convergence if we used a different test? 2. How does the factorial in the denominator influence the convergence of the series? 3. What is the significance of the alternating factor $$(-1)^n$$ in this series? 4. How would the series change if we had $$(2n+1)!$$ in the denominator instead? 5. What if $$x$$ were replaced by a complex variable? How would convergence be affected? **Tip:** The Ratio Test is particularly useful for series with factorials or exponential terms due to its reliance on the limit of the ratio between terms.

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Math Problem Analysis

Mathematical Concepts

Calculus
Series
Convergence
Ratio Test

Formulas

Radius of convergence using Ratio Test: \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)
General form of a series: \( \sum a_n \)

Theorems

Ratio Test for Convergence

Suitable Grade Level

Grades 11-12, College Level

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