The following problem comes from a 1961 exam set collected by Ed Barbeau of the University of Toronto. The discontinued exams (by 2003) were for 5^{th} year Ontario high school students seeking entrance and scholarships for the second year at a university.

“If *s*_{n} denotes the sum of the first *n* natural numbers, find the sum of the infinite series

.”

Unfortunately, the “Grade XIII” exam problem sets were not provided with answers, so I have no confirmation for my result. There may be a cunning way to manipulate the series to get a solution, but I could not see it off-hand. So I employed my tried and true power series approach to get my answer. It turned out to be power series manipulations on steroids, so there must be a simpler solution that does not use calculus. I assume the exams were timed exams, so I am not sure how a harried student could come up with a quick solution. I would appreciate any insights into this.

Answer.

See Serious Series for a solution.

(**Update 1/18/2021**) Another Solution Continue reading →