What's the difference between radius of convergence and interval of convergence?

Question

Emilia Aguilar · Accepted Answer

As an illustration, consider the following: the interval of convergence provides the precise values of where the series converges and does not, whereas the radius of convergence gives us the number of values where the series converges.

#sum_(n = 1)^oo(2^n (x+ 2)^n)/((n + 2)!)#

We use the ratio test to find our radius of convergence. We know that the ratio states that if #lim(n-> oo) a_(n + 1)/a_n < 1#, the series converges. We want to find the values of #x# where the series converges. Thus

#lim_(n-> oo) ((2^(n + 1)(x + 2)^(n + 1))/((n + 1 + 2)!))/((2^n(x +2)^n)/((n + 2)!)) < 1#

#2(x+ 2) lim_(n-> oo) 1/(n + 3) < 1#

#0 < 1#

Since this is true for all values of #x#, the radius of convergence is #oo# and therefore the interval is #(-oo, oo)#.

I hope this is useful!

Elijah Abram · Answer

undefined The radius of convergence of a power series is a non-negative real number that determines the convergence behavior of the series. It represents the distance from the center of the series to the nearest point where the series converges. The interval of convergence, on the other hand, is the set of all real numbers for which the power series converges. It consists of all the values of the variable for which the series converges, including the endpoints of the interval if applicable. The interval of convergence may be an open interval, a closed interval, or a half-open interval depending on the convergence behavior of the series at its endpoints.