II. Sequence Convergence Problem

A term-by-term analysis of a simple series for computing ln(2), the natural logarithm of 2, concludes that it converges very slowly. You will compare two formulas for computing ln(2).

Consider a formula for the natural logarithm

ln(1+x) == x - x2/2 + x3/3 - x4/4 + x5/5 - ...

(The proof is by calculus; integrate the power series expansion of 1/(1+x).)

Define the sequence of partial sums by:

S = { x, x - x2/2, x - x2/2 + x3/3, ..}

Part 1 of this problem is to compute and display numeric values of the first four terms of this sequence for x = 1. Show 5 decimal places for each number. You will see very slow convergence to the true value of ln(2), viz. 0.69315.

A better power series for computing natural logarithms is easily derivable from this one. We note that, by well-known properties of the logarithm,

ln((1+x)/(1-x)) == ln(1+x) - ln(1-x)

Using the series for ln(1+x) above, and replacing x with -x to get a series for ln(1-x), we subtract the two series and conclude that

ln[(1+x)/(1-x)] == 2 [ x + x3/3 + x5/5 + x7/7 + x9/9 ... ]

Setting x = 1/3, the above formula will also compute ln(2).

Part 2 is to compute the first four partial sums from this formula to 5 decimal places, and display them. Note how much faster it converges.

Part 3: the formula for ln(1-x) indicated above can also be used to compute ln(2). How fast do the partial sums converge for this formula?