# geometric series

 What is 1-1+1-1+1-1+...? How infinite sums challenge our notion of arithmetic. When you flip a coin we assume it has equal chance of coming up head or tails, so any coin flipping game should be a fair one. But Yutaka Nishiyama and Steve Humble can give you the winning advantage. Infinite series occupy a central and important place in mathematics. C. J. Sangwin shows us how eighteenth-century mathematician Leonhard Euler solved one of the foremost infinite series problems of his day. The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides. The harmonic series is far less widely known than the arithmetic and geometric series. However, it is linked to a good deal of fascinating mathematics, some challenging Olympiad problems, several surprising applications, and even a famous unsolved problem. John Webb applies some divergent thinking, taking in the weather, traffic flow and card shuffling along the way. Those who understand compound interest are destined to collect it. Those who don't are doomed to pay it - or so says a well-known source of financial advice. But what is compound interest, and why is it so important? John H. Webb explains.