Harrell's little crib sheet on series - examples

© Copyright 2000 by Evans M. Harrell II

  1. (geometric series) Suppose you are paying off a college loan with initial value p0, interest rate per period r, and payment m. At the end of payment period n, the amount you owe is pn = pn-1 (1+r) - m. Use the geometric series to solve for pn in terms of p0, r, and m. When is the loan paid off?
  2. (p-series) Put a unit mass on every integer point of the positive x-axis. Will the gravitational force on a particle, such as yourself, standing at the origin be finite or infinite? (Gravity is an inverse-square force.) Variants: a) Put mass sin2(n) at x=n; b) Put mass n at x=n; c) Put mass n1/2 at x=n.
  3. Use your knowledge of the derivative of the arctan(x) to find its Taylor series painlessly. Use the result to evaluate the series
  4. (convergence by Taylor) Use the ratio test to verify that converges. Then use Taylor series and some ingenuity to find the limit.
  5. (hitting with calculus) Integrate the geometric series in x term by term to get another Taylor series. To what function does it converge? For what x does it converge?
  6. (bigger than an elephant) diverges, because at least for n > 2, and the first million terms don't matter. So this series is bigger than half an arithmetic series, which diverges.
  7. (lack of commitment) The series can't make up his mind whether to settle down with with 0 or 1. The more adventuresome students can think about
  8. (ratio with > 1) Explain why the ratio test for divergent series is equivalent to the comparison test with a diverging geometric series.
  9. (smaller than a bug) Use comparison to show that converges. Could any other convergence tests also squash this series?
  10. (ratio test) Explain why the ratio test for convergent series is equivalent to the comparison test with a converging geometric series.
  11. (ratio test) Tip: This is a good test to apply if there are terms like n! and cn in an (examples).
  12. (alternating series) Consider the alternating sum of the reciprocals of the odd integers, . Use your calculator to get an impression of how rapidly it converges. Use the Taylor series for the arctangent , above, to find its sum. Discuss what happens if it is summed in other orders.
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