Computing Natural Logarithm of 2

[JeromePl]

I got this question and I know a bit on how to start it, but not sure which direction is best:

By considering the power series 1/(1 + x) and 1/(1 - x) show that:

so I do the differentiation - ln(1 + x) = ^x₀ du/1 + u = x - x²/2 + x³/3 - x⁴/4 ...

which equals - = sigma (upper infinity and lower is k=0) (-1)^k x^k+1/k+1

and for ln(1 - x) I do the same...

but how do I show this on MATLAB and what else do I need to do?

The next question is then:

Hence show that ln (1+x/1-x) = 2 (x + x³/3 + x⁵/5 + x⁷/7...) = 2 sigma (upper is infinity and lower is k = 0) x^{2k +1}/2k +1

then is says determine the range of values of x for which each of the above series will converge.




Next question:

Define S_n(x) and Si_n(x) to be partial sums for the infinite series above:

S_n(x) = Sigma (upper is n and lower k = 0 ) (-1)^k x^k+1/K+1

S'_n(x) = 2 Sigma (n is upper and lower k=0) x^2k+1/2k +1

for the above write two MATLAB functions. Using suitable values for x construct a table of estimates for ln(2) and the errors E_n = abs(S_n - ln(2)) and E'_n = abs(S'_n - ln(2)).

using the table of data, estimate the rate of convergence for the first series S_n.

the estimate how many terms would be needed in each series to ensure an accuracy of 5 decimal places.

 

[Valkarie]

To solve this question, you will need to first define the two functions Sn(x) and S'n(x) in MATLAB. To do this, you can use a for loop to calculate the partial sums of both series. You can then construct a table of estimates for ln(2) using suitable values for x, and the errors En and E'n. Finally, you can use the data from the table to estimate the rate of convergence for the first series Sn, as well as how many terms would be needed in each series to ensure an accuracy of 5 decimal places.