Evaluate $\sum_{r = 2}^{n} ln (\frac{r ^{2} - 1}{r ^{2}})$ in terms of $n$ .

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Find the Maclaurin expansion of $f (x) = ln (1 + 2 x)$ up to and including the term in $x^{3}$ .
Evaluate the sum to infinity: $\sum_{r = 1}^{\infty} \frac{1}{r ( r + 1 )}$ .
An arithmetic progression has first term $a = 5$ and common difference $d = 3$ . If the sum of the first $n$ terms is $S_{n} = 260$ , find $n$ .
Find the range of values of $x$ for which the geometric series $1 + (2 x - 1) + (2 x - 1)^{2} + \dots$ converges.
Using the series expansions for $e^{x}$ and $cos x$ , find the first two non-zero terms in the Maclaurin series for $e^{x} cos x$ .
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