Exercises

In 1-6, determine the interval of convergence for each power series. You do not have to determine whether or not the series converge at the endpoints.

In 7-12, compute the limits using the technique shown in Example 5.

• Suppose a power series converges for and diverges for all other values of x. What is the value of and why?
• Suppose that a power series converges for -6<x<6. Will the power series converge faster for x=2 or for x=3? Explain your reasoning.
• Consider .
  1. Compute the radius of convergence for f(x).
  2. Differentiate term by term to find the series for f'(x), and then find its radius of convergence.
  3. Let F(x) be the function such that F'(x)=f(x) and F(0) = 5. Find the power series expansion for F(x) (around x=0), and find its radius of convergence.
• Let .
  1. The series for f(2) converges like a geometric series. What is the ratio for this geometric series?
  2. Using 15 terms from the sum for f(2) we think we have an error of less than 0.1. How many terms should we sum to get an error less than ?
  3. Is your estimate on the number of terms in part (b) likely to be too large or too small? Explain your reasoning.
• Let .
  1. The series for f(3) converges like a geometric series. What is the ratio for this geometric series?
  2. Using 10 terms from the sum for f(3) we think we have an error of less than 0.001. How many terms should we sum to get an error less than ?
  3. Is your estimate on the number of terms in part (b) likely to be too large or too small? Explain your reasoning.