Chapter 9.2, Problem 79E

Probability A fair coin is tossed repeatedly. The probability that the first head occurs on the nth toss is given by $P(n)={\left(\frac{1}{2}\right)}^n$, where $n\ge 1$.

(a) Show that ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{\left(\frac{1}{2}\right)}^n=1}$

(b) The expected number of tosses required until the first head occurs in the experiment is given by

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}n{\left(\frac{1}{2}\right)}^n}$

Is this series geometric?

(c) Use a computer algebra system to find the sum in part (b).