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2 problems match “infinite series”. Clear

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1. Probability the game never stops ●●○○○ 20 ptsA box starts with one red ball and one blue ball. At each step, a ball is taken at random from the box. If it is red, the game stops. If it is blue, it is returned to the box together with one additional blue ball. Thus, at the $n$-th draw the box contains $n+1$ balls, one of which is red. Find the probability that the game never stops.
2. Arithmetic-geometric series in complex z ●●○○○ 20 ptsLet $z=1-2i$. Find the real part and the imaginary part of $\frac{1}{z}+\frac{2}{z^{2}}+\frac{3}{z^{3}}+\cdots$.