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

so I do the differentiation - ln(1 + x) =

^{x}

_{0}du/1 + u = x - x

^{2}/2 + x

^{3}/3 - x

^{4}/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 + x

^{3}/3 + x

^{5}/5 + x

^{7}/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.