Explain the purpose of the Ratio Test in determining convergence of a power series.
The Ratio Test helps determine the radius of convergence by finding the values of x for which the limit L < 1.
Why is it necessary to test the endpoints of the interval obtained from the Ratio Test?
The Ratio Test only provides an open interval. Endpoints must be tested individually to determine if they are included in the interval of convergence.
What does the radius of convergence tell you?
The radius of convergence tells you how far from the center the power series converges.
How do power series relate to functions?
Power series can represent functions over an appropriate interval. They are approximations of functions.
What happens when L = 1 in the Ratio Test?
The Ratio Test is inconclusive. Another test must be used to determine convergence or divergence.
What is the general form of a power series?
$\sum^{\infty}_{n=0}a_n(x-r)^n$
Write the formula for the Ratio Test.
$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$
What condition on L in the Ratio Test implies convergence?
$L < 1$
What condition on L in the Ratio Test implies divergence?
$L > 1$
How do you find the radius of convergence of a power series?
1. Apply the Ratio Test. 2. Solve the inequality L < 1 for |x - r|. 3. The radius of convergence, R, is the value such that |x - r| < R.
How do you determine the interval of convergence of a power series?
1. Find the radius of convergence, R. 2. Test the endpoints x = r - R and x = r + R in the original series. 3. Determine whether each endpoint converges or diverges. 4. Write the interval of convergence using appropriate brackets.
What is the first step in finding the interval of convergence for $\sum^{\infty}_{n=0}\frac{2^n}{n}(4x-8)^n$?
Apply the Ratio Test to the series.
After applying the Ratio Test, how do you find the radius of convergence?
Set the limit L < 1 and solve for |x - r|. The value on the right side of the inequality is the radius of convergence.
After finding the radius of convergence, what's the next step to find the interval of convergence?
Test the endpoints of the interval by plugging them back into the original series and determining if they converge or diverge.
How do you test the endpoints of the interval of convergence?
Substitute each endpoint value into the original power series. Then, analyze the resulting series using convergence tests (e.g., alternating series test, p-series test).
