Radius of convergence

Question

If the interval of convergence is (-2,6), then what is the radius of convergence,R? Round to one decimal place.

Answer 1

Ruby, I hope you understand the "big idea" here-- the interval of convergence is the "list" of all x- values where the power series gives the same results as the function.

For example, f(x) = 6/(1 - x) = 6 + 6x + 6x²+ ... but only for x values (-1,1)

I recommend that you graph the function and some terms of the power series to see this-- the graphs only " match up" or coincide for these values of "x." If we let x = 4, the power series is infinitely large and the function is 6/-3 = -2-- these are very different results because we used a value outside of the interval of convergence.

If your interval of convergence is (10,18), then you are saying those values of "x" are where the power series is equivalent to the given function, and the radius of convergence is 4

The radius of convergence is half of the interval of convergence, and you can imagine a circle with a diameter of the interval of convergence and the ask what the radius of this circle is. We call it a radius because we will have to have a picture of a sphere of convergence when we move to doing this in 3-dimensions in Multi-variable Calculus, but you don't have to work with that yet.

Hope this helps, but please let me know.--Andrew