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A power series is a way to represent a function as an infinite sum of terms, each of which is a power of the variable multiplied by a coefficient. The general form of a power series for a function is given by:
\[ f(x) = \sum_{n=0}^{\infty}a_nx^n \].
In this expression, \( a_n \) are the coefficients of the series and 'n' is the index that runs from 0 to infinity. Each term in the series involves a power of 'x' and the corresponding coefficient.
Power series are incredibly useful because they allow us to work with very complicated functions in a much simpler, polynomial-like form.

• Example: The exponential function \( e^x \) can be expanded as a power series: \[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \].

Using power series, we can solve differential equations and find solutions that might otherwise be very difficult to obtain.

When dealing with power series, one important concept is the 'radius of convergence'. This term refers to the distance within which the power series converges to a finite value as you move away from the center of the series (usually around \( x=0 \)).
The radius of convergence helps us understand where the series representation of a function is valid. To find this radius, we often use the ratio test. For a power series \( \sum_{n=0}^{\infty} a_n x^n \), the ratio test involves finding:
\[ \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \].

• If this limit is 'L', then the radius of convergence 'R' is given by \( R=\frac{1}{L} \).
• If \( L=0 \), the series converges everywhere, thus having an infinite radius of convergence.
• If \( L \to \infty \), the series converges only at the center point.

In our specific problem, the limit goes to 0 as \( n \to \infty \), implying that the radius of convergence is infinite.

An analytic solution of a differential equation is one that can be expressed in terms of power series. We call a function 'analytic' at a point if it can be represented by a convergent power series in some interval around that point. Let's break down why this is important:

• Exact Results: Analytic solutions give us exact results within the radius of convergence, unlike numerical methods which provide approximate results.
• Explanation of Behavior: Power series solutions can provide insight into the behavior of the function near specific points.

For the given differential equation \( f''(x) = -f(x) \), we found an analytic solution using a power series. We assumed:\[ f(x) = \sum_{n=0}^{\infty} a_n x^n \] and derived the coefficients. The resulting series, \( f(x) = x - \frac{x^3}{6} + \frac{x^5}{120} - \ldots \), is an exact analytic solution representing the function for all real values of x, as the radius of convergence is infinite.