Speedy Series for Lethargic Logarithms

(This project was adapted from Student Research Projects in Calculus, M. Cohen et. al., MAA, 1991.)

This project may be done in a group of up to 4 people. Each member of the group will receive the same grade. The project is due Tuesday, April 23.

Many handbooks of mathematics list several infinite series expansions for the natural logarithm ln(x). For example, CRC Standard Mathematical Tables and Formulae (30^th edition), which is in the MacArthur library, lists 4 different series for ln(x) (which it denotes log(x)). While these books give the radius of convergence for each series, they do not address the question of which series converges the most quickly - a vital question if you are trying to actually compute an approximation in a reasonable amount of time. We are going to compare two of these series and address this question.

Part 1: First Series for ln(x)

To begin with, we will look at the usual Taylor series for ln(x), expanded about x = 1.

1. Compute the Taylor series for ln(x) expanded about x = 1 (you must show your computation, not just quote the result in the text!).

2. Find the interval of convergence for this series. Be sure to check the endpoints!

3. Compute all of the partial sums, for up to ten terms, of ln(0.5) and ln(2), using this series. What are the errors after 10 terms (compared to the values given by your calculator)?

4. Determine how many terms you would need to compute to have 10 decimal places of accuracy when x = 0.1, 0.5, 1, 2. You may assume that, when the series converges, it converges to ln(x). [Hint: Compare the error term in each case to an appropriate geometric series.]

5. How could you use this series to compute ln(10)?

Part 2: Second Series for ln(x)

Now we will look at a different series for ln(x). Consider the transformation of variables:

1. Compute the Taylor series for , in terms of the variable y, expanded about y = 0.

2. Find the interval of convergence for this series in terms of y. Be sure to check the endpoints!

3. Find the interval of convergence for the series in terms of x. How does this compare to the interval of convergence for our first series?

4. Compute all of the partial sums, for up to ten terms, of ln(0.5) and ln(2), using this series. What are the errors after 10 terms (compared to the values given by your calculator)?

5. Determine how many terms you would need to compute to have 10 decimal places of accuracy when x = 0.1, 0.5, 1, 2, 10. You may assume that, when the series converges, it converges to ln(x).

Part 3: Comparing the Series

1. Compare the results from parts 1 and 2 by constructing appropriate tables which summarize your calculations. What are your conclusions?

Let y = t(x) be the inverse of the transformation .

2. Find t(x). Show that .

3. How could you use this relationship to avoid computing the series in part 2 for negative values of y?

4. Is there any value in avoiding negative values of y? Compare with the analogous situation for the series in part 1 (consider your answer to question 5 in part 1).