Radius and Interval of Convergence
In this example, we demonstrate how to find the radius and interval of convergence of a power series sum of (-1)nxn/((n+2)2n). There are two parts to the process when done by hand: part 1, apply the Ratio Test to find the radius of convergence:
1) Compute |an+1/an| (Python command subs)
2) Simplify and take the limit as n->oo (Python commands simplify and limit)
3) If it is not intuitively obvious, solve the inequality L<1 (Python command: reduce_inequalities)
We begin by importing the sympy package and defining n and x as symbolic variables. We list them in separate commands since we want n to be a positive integer, but not x. In step 1, we define a and use the subs command to get an+1. In step 2, we simplify the expression (printing it to check our progress), then computing the limit as n approaches infinity (using “oo” for infinity). It is clear from our limit of |x|/2 that x must be between -2 and 2. If it wasn’t obvious, we can use the reduce_inequalities command to solve L<1. For part 2, we must test the endpoints by substituting into a (subs), simplifying the result (simplify), and determining whether the resulting series converges or not (by inspection is OK if possible). When x=-2, the terms of the series are 1/(n+2). This series diverges by Limit Comparison with the sum of 1/n. When x=2, the terms of the series are (-1)n/(n+2), which converges by the AST.