Exercise on infinite series, Friday, October 28, 2005, Math 172-502

The goal of this exercise is to increase your understanding of convergence of infinite series by exploring some examples both theoretically (by hand) and experimentally (by computer).

Section 10.3 in the textbook contains the background information about the integral test, the comparison test, and the remainder estimate for the integral test. The relevant Maple commands are add(f(n),n=1..10) (for the sum of a finite number of terms), sum(f(n),n=1..infinity) (for the sum of a series), and evalf (for floating-point evaluation).

Three questions

Answer the following questions for each of the expressions for an indicated below.

  1. Use Maple to compute the sum of the first 10 terms; the first 100 terms; the first 1000 terms. (Assume the sum starts with the n=1 term.) What seems to be happening as you add more terms?
  2. Does the infinite series Σan converge or diverge? Can you prove it?
  3. If the series converges, can you determine the sum to 15 decimal-place accuracy? Can you predict roughly how many terms need to be added to compute the sum to that accuracy?

Four examples

  1. an=1/(1+n4)
  2. an=n*exp(-n2)
  3. an=1/(n*ln(n+1))
  4. an=ln(n)/n2