## Thursday, January 1, 2015

### Limit vs. Limit graphs to analyze infinite series

f(x) = 1/x^3 as x goes to infinity will reach 0, and the integral from 1 to infinity will reach 1/2. I find it interesting to parameterize a 2 variable curve where one variable s is the function's value at t and the other r is the partial integral from 1 up to t, and make a plot of s vs. r like below.
This shows the behavior of f(x) over the entire x axis from 1 to infinity as it relates to the integral to that point.

It leads to an interesting proof of the infinite sum of geometric series... Here I plot the sum of 1/2^n up to t vs. the value of each individual term at t...
The value of the first term is 1 and the sum of just that term is of course 1, and the next term is 1/2 and the sum of the first two terms is 1.5, and it precedes linearly all the way to a term being 0 and the sum of the infinite terms being 2. But if you know all the points lie on the same line you can use just the first two terms to derive a line that shows ultimately the infinite sum will reach the r axis at 1/(1-1/2). And generalize that to any geometric series.