Tag Archives: infinite series

A Nice Factorial Sum

This is another infinite series from Presh Talwalkar, but with a twist.

“This problem is adapted from one given in an annual national math competition exam in France. Evaluate the infinite series:

1/2! + 2/3! + 3/4! + …”

The twist is that Talwalkar provides three solutions, illustrating three different techniques that I in fact have used before in series and sequence problems.  But this time I actually found a simpler solution that avoids all these.  You also need to remember what a factorial is: n! =n(n – 1)(n – 2)…3·2·1.

See a Nice Factorial Sum

Another Christmas Tree Puzzle

This is a belated Christmas puzzle from December 2019 MathsMonday.

“A Christmas tree is made by stacking successively smaller cones. The largest cone has a base of radius 1 unit and a height of 2 units. Each smaller cone has a radius 3/4 of the previous cone and a height 3/4 of the previous cone. Its base overlaps the previous cone, sitting at a height 3/4 of the way up the previous cone.

What are the dimensions of the smallest cone, by volume, that will contain the whole tree for any number of cones?”

Recall that the volume of a cone is π r2 h/3.

See Another Christmas Tree Puzzle

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 Continue reading

Number of the Beast

If you will pardon the pun, this is a diabolical problem from the collection Five Hundred Mathematical Challenges.

Problem 5. Calculate the sum

__________

It has a non-calculus solution, but that involves a bunch of manipulations that were not that evident to me, or at least I doubt if I could have come up with them. I was able to reframe the problem using one of my favorite approaches, power series (or polynomials). The calculations are a bit hairy in any case, but I was impressed that my method worked at all.

See the Number of the Beast