91Ó°ÊÓ

When examining a power series like \( \sum_{k=0}^{\infty} \frac{k !}{2^{k}} x^{k} \), it is crucial to determine where this series will converge. One useful tool for this is the ratio test.
The ratio test helps us figure out whether a series converges by comparing the limit of consecutive terms.
For a power series of the form \( \sum a_k x^k \), the ratio test states that the series will converge if:

• \( \lim_{k \to \infty} \left | \frac{a_{k+1}}{a_k} x \right | < 1 \)

Applying this to our series means calculating \( \lim_{k \to \infty} \left | \frac{(k+1)!}{2^{k+1}} \cdot \frac{2^k}{k!} \right | \cdot |x| \).
After substitution and simplification, the ratio becomes \( \left(\frac{|x|}{2} \right) (k+1)\).
As \( k \) approaches infinity, the expression goes to infinity unless \( x = 0 \). This means the series will converge only at that point.

The interval of convergence tells us the specific values of \( x \) for which a power series converges.
Once we know the radius of convergence, finding this interval becomes easier.
In our example, after using the ratio test, the series only converges when \( x = 0 \), meaning the radius of convergence is \( R = 0 \).
The interval of convergence is then simply the single point \( x = 0 \).
This means the power series does not converge for any other values of \( x \), because the ratio test indicated divergence elsewhere.
Thus, understanding the radius of convergence directly tells us the limited interval where the series is convergent.

A power series is a series of the form \( \sum_{k=0}^{\infty} a_k x^k \), where each term is a power of \( x \) multiplied by a coefficient \( a_k \).
Power series are important in mathematics because they can represent functions over certain intervals.

• They offer ways to approximate more complex functions.
• Many functions can be expressed as a power series.

In our specific problem, the series \( \sum_{k=0}^{\infty} \frac{k !}{2^{k}} x^{k} \) is a power series centered around \( x = 0 \).
We describe a power series in terms of its coefficients \( a_k \) and radius of convergence.
The radius is crucial as it tells us the extent to which values of \( x \) in the real number line this series converges.
Power series can also help in solving differential equations and modeling physical phenomena.