Question

Determine the radius of convergence of the given power series. $$ \sum_{n=0}^{\infty} \frac{x^{2 n}}{n !} $$

Short answer

Answer: The radius of convergence of the given power series is ∞.

Step-by-step solution

Apply the Ratio Test

Applying the ratio test, we need to find the limit: $$ \lim_{n\to\infty} \frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \frac{\frac{x^{2(n+1)}}{(n+1)!}}{\frac{x^{2n}}{n!}} $$

Simplify the Ratio

Let's simplify the ratio before finding the limit: $$ \lim_{n\to\infty} \frac{x^{2(n+1)}}{(n+1)!} \cdot \frac{n!}{x^{2n}} = \lim_{n\to\infty} \frac{x^{2n+2}}{(n+1)!} \cdot \frac{n!}{x^{2n}} $$ Now, we can cancel \(x^{2n}\) and \(n!\) in the numerator and denominator: $$ \lim_{n\to\infty} \frac{x^2}{(n+1)!} \cdot \frac{n!}{1} = \lim_{n\to\infty} \frac{x^2 n!}{(n+1)!} $$

Compute the Limit

Next, we compute the limit as \(n \to \infty\): $$ \lim_{n\to\infty} \frac{x^2 n!}{(n+1)!} = \lim_{n\to\infty} \frac{x^2 n!}{(n+1)n!} $$ We can cancel out \(n!\) in the numerator and denominator: $$ \lim_{n\to\infty} \frac{x^2}{(n+1)} = 0 $$ As the limit is zero for any value of x, the ratio test always gives a value less than 1, which means the power series converges for all x.

Conclusion

The radius of convergence is given as: $$ R = \lim_{n\to\infty} \frac{1}{\lim_{n\to\infty} \frac{x^2}{(n+1)}} = \lim_{n\to\infty} (n+1) = \infty $$ Thus, the radius of convergence of the given power series is \(\infty\), and the series converges for all values of x.

Key concepts

Ratio Test

The ratio test is a powerful tool for determining the convergence of an infinite series. It's particularly handy when dealing with power series. To apply the ratio test, we examine the limit of the ratio of successive terms in the series. More formally, given a series \( \sum a_n \), we consider \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

For the power series in question, the ratio test helps us see under what conditions the series converges. It's a straightforward method that gives us quick insight into the behavior of infinite series.

Power Series

A power series is specifically a series of the form \( \sum_{n=0}^{\infty} c_n (x-a)^n \), where \( c_n \) represents the coefficients and \( a \) is the center of the series. In simpler terms, it's like a polynomial but with infinitely many terms. Power series are widely used in mathematics due to their flexibility in representing functions.
In our exercise, we are dealing with the power series \( \sum \frac{x^{2n}}{n!} \). Here:

• Each term is of the form \( x^{2n} \), indicating that the series is centered at zero.
• We use factorial \( n! \) in the denominator, which impacts the convergence behavior significantly.

Power series are central to many areas of calculus and analysis and can model a variety of functions, expanding in powers of \( (x-a) \).

Convergence

Convergence in the context of a series refers to whether the series approaches a finite value as the number of terms goes to infinity. For a series \( \sum a_n \) to converge, the sequence of its partial sums, \( S_n = \sum_{k=0}^{n} a_k \), must approach a specific number.
For power series, you need to determine a range of \( x \) for which the series converges. This is tied to the radius of convergence. With our given series, the convergence needs to be assessed using the ratio test.Using the ratio test, we find that the series \( \sum \frac{x^{2n}}{n!} \) converges for all \( x \), because the test reveals that the limit goes to zero for any \( x \). Hence, we can say that the entire series converges across the complex plane.

Infinite Series

An infinite series is the sum of an infinite sequence of terms. It's symbolized as \( \sum_{n=0}^{\infty} a_n \). Unlike finite sums, which yield a concrete result, infinite series don't always produce a clear sum.
They require special convergence tests, like the ratio test, to inform us whether they have a finite limit.Infinite series are a fundamental concept in analysis and are used to model and approximate complex systems and functions. The infinite series in our problem expresses a sequence of powers of \( x \) divided by factorial terms. Its form allows us to explore the behavior of exponential functions and solutions to differential equations, thanks to its convergence properties. The fascinating part about infinite series is that while they seem endless, under the right conditions, their sum can be finite or infinite, giving rise to essential insights in mathematics.