(a) Write the following series with a sum sign and compute them using the geometric series.
\( \begin{array}{cc} 3+r^{3}+r^{4}+r^{5}+\cdots & 0 . \overline{17}= & \frac{1}{10}+\frac{7}{10^{2}}+\frac{1}{10^{3}}+\frac{7}{10^{4}}+\cdots \\ 1-s+s^{2}-s^{3} \pm \cdots & 1-x^{2}+x^{4}-x^{6} \pm \cdots \end{array} \)
(b) Find the value of the series
\( 1+2 u+3 u^{2}+4 u^{3}+\cdots \)
Hint: Compute \( S-u S \), where \( S \) is the wanted value. Doing this with an infinite series is only allowed, if one knows in advance that the series is convergent - which is true for \( |u|<1 \) in the above case.
