The first power series above always yields a geometric series with ratio equal to . Based on what we know about geometric series, we can immediately conclude that for values of between and 1, the power series will sum to . For values of with absolute value bigger than or equal to 1, the power series will diverge. Thus, if we set , then it turns out that Here's the Maple command to subs titute a different value into an expression, and an illustration that we always get geometric series using the example above. I'll define to be the expression for this power series.
Some values yield convergent series, some yield series which diverge.
Determining which values lead to convergent series, and which lead to divergent series is the main topic of this lab. The key will be to examine this question for all values of simultaneously. All series converge if and only if the sequence of partial sums converge, that is the official definition. For power series, partial sums are just regular polynomials. We'll look at these partial sums graphically, to illustrate certain properties, then apply the ratio test.