Let x be the sum of the values of sin(n) for the whole numbers 0 though n. (In radians, of course.)
Find the maximum A and minimum B such that
A < x < B
Adapted from AMS Page A Day Calendar by Evelyn Lamb December 7.
The AMS calendar (which I recommend, it has dates but not days of the week so its reusable) only gives conservative bounds of -1/6 < x < B. I though these might be provable exact bounds so I submitted this as a problem then tried working on it.
After trying the problem I realized it would probably become open ended. So I may as well share what I have.
What I noticed is the numbers for n that give large and small sums.
For example when n=12 we get one of the first very small sums -.12533
This comes not just from the fact that there are 3 negative terms in a row: sin(10), sin(11), sin(12) but from the middle term being almost exactly -1. sin(11)=-.9999902066 So the three terms are all quite small. This in turn comes from 11 radians being nearly a quadrantal angle. 11 = 3.5 rotations gives
pi = 11/3.5 = 22/7
Other good rational approximations of pi that can be written as P/(Q+.5) for integers P,Q will allow us to find other high and low sums.
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Posted by Jer
on 2020-12-11 12:43:29 |
My take
