Serious Series

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 5th year Ontario high school students seeking entrance and scholarships for the second year at a university.

“If sn 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 7th 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 7th 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.

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