Comments on: Closer to One

By: Justin Lanier

Justin Lanier — Thu, 24 Apr 2014 18:41:43 +0000

@mrdardy: Your way makes me think to write n/(n-1) as n/(n-1) – 1/(n-1) + 1/(n-1) = 1 + 1/(n-1). This might be useful in comparing with (n-1)/n = 1 – 1/n.

By: Justin Lanier

Justin Lanier — Thu, 24 Apr 2014 18:34:27 +0000

I thought: (n-1)/n and n/(n-1) are each fractions that are one “piece” away from being 1. You’re closer to 1 when the pieces are smaller, and hence when the denominator is “larger”. At first I was just thinking about positive numbers, where (n-1)/n is one piece less than 1 and n/(n-1) is one piece more than one. And so n-1/n, by virtue of its smaller pieces, is closer to 1. But then I thought about negative numbers, and realized that the roles would be reversed. That’s how I knew that it depends.

Later on I thought about “cross multiplying” and comparing n^2 and (n-1)^2. Clearly which is larger depends on n.

By: mrdardy

mrdardy — Wed, 23 Apr 2014 19:12:15 +0000

Michael
My first thought (as a Calculus teacher) is that the first expression is 1 – 1/n and as n gets large this is very close to 1. The second expression can be rewritten (by long division of n/(-1+n)) as -n – n^2 – n^3 … and the graph of the rational function shows that this also nears 1 as n gets large. It also appears that the first expression converges more quickly. So, my next idea would be to simply substitute a couple of manageable n values into these expressions to get a table of values going. Looking at n =1 to n = 8 the first expression is closer for each value. Fun problem to think about. I may toss this out to my BC kids and see what they do with it!