- (geometric series) Suppose you are paying off a college loan with
initial
value p
_{0}, interest rate per period r, and payment m. At the end of payment period n, the amount you owe is p_{n}= p_{n-1}(1+r) - m. Use the geometric series to solve for p_{n}in terms of p_{0}, r, and m. When is the loan paid off? - (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 sin
^{2}(n) at x=n; b) Put mass n at x=n; c) Put mass n^{1/2}at x=n. - Use your knowledge of the derivative of the arctan(x) to find its Taylor series painlessly. Use the result to evaluate the series
- (convergence by Taylor) Use the ratio test to verify that converges. Then use Taylor series and some ingenuity to find the limit.
- (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?
- (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.
- (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
- (ratio with > 1) Explain why the ratio test for divergent series is equivalent to the comparison test with a diverging geometric series.
- (smaller than a bug) Use comparison to show that converges. Could any other convergence tests also squash this series?
- (ratio test) Explain why the ratio test for convergent series is equivalent to the comparison test with a converging geometric series.
- (ratio test) Tip: This is a good test to apply if there are terms
like n! and c
^{n}in a_{n}(examples). - (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.

Return to crib sheet