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.

See Serious Series

(**Update 1/18/2021**) Another Solution

I received another solution from Evander Tandiarrang (of Hard Geometric Problem fame) from Papua New Guinea, now in 7^{th} grade! It looks correct to me and only relies on the geometric series without explicit calculus. Needless to say, I was nowhere near such mathematical prowess in 7^{th} grade.

Pada Senin, 18 Januari 2021 11.06.52 GMT+9

Good morning and Happy New Year,

Every time I practice with mathematic and science problems, and now I try to solved the Serious Series problem, but may be not correct. Please give me a correction for my solution.

Regards,

Evander T.