180 Days of Ideas for Discussion in Math Class.
(as of 9July2014, we're in overtime!)
Quick, which infinite sum is greater?
1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 ...,
1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + 13/128 ...
— Matt Enlow (@CmonMattTHINK) April 29, 2014
Dang, I lost my comment. Intriguing. I thought about convergence, and convinced myself that both converge. But I didn't know either sum. wolframalpha's answer intrigued me. I'll keep thinking about this.
For a method of evaluating these sums, see the end of my answer athttp://math.stackexchange.com/questions/766798/what-are-power-series-used-for-a-reference-requestSpecifically, the problem with parts labeled 4(a) through 4(f). This method makes use of calculus methods. Here's a way of finding an explicit closed form expression for the sum using only "high school algebra" manipulations:http://mathforum.org/kb/message.jspa?messageID=6326740See also the other posts in the above sci.math thread archived at Math Forum.
The thing is, the question asked which is bigger, not whether they'd converge. Once you get past the terms listed, it's should become obvious to the students that, term for term, the one is smaller than the other.If the concern is that divergent series can't be compared for size, then maybe we should have asked for a comparison of the first billion terms instead.