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Unformatted text preview: MATH 118, LECTURE 26: POWER SERIES 1 Power Series A few weeks ago when considering the convergence of geometric series we encountered the example ∞ summationdisplay n =1 parenleftbigg 2 3 parenrightbigg n x n- 1 . We considered the convergence as we did for normal numerical geometric series and found that the series converged in the finite range- 3 / 2 < x < 3 / 2 so that ∞ summationdisplay n =1 parenleftbigg 2 3 parenrightbigg n x n- 1 = 2 3- 2 x ,- 3 2 < x < 3 2 . At the time, we did not give any more thought to the example. It was sufficient for us to see that, for any value of x in the range, the numerical se- ries converged. (For instance, if we chose x = 1 we have ∑ ∞ n =1 (2 / 3) n x n- 1 = ∑ ∞ n =1 (2 / 3) n which converges to 2 / (3- 2 x ) = 2 / (3- 2) = 2, etc.) However, there is something far more interesting happening here. The original series is a series of functions and the limit of convergence is a func- tion as well. So we have a series of functions converging to a function on a certain interval. What could this possibly mean? Let’s consider the first few partial sums in the series ∑ ∞ n =1 (2 / 3) n x n- 1 given by P n- 1 ( x ) = n summationdisplay k =1 parenleftbigg 2 3 parenrightbigg k x k- 1 . If we compare a few of these functions to the limiting function 2 / (3- 2 x ) we can see that the sequence of partial sums becomes a better and better approximate of the function f ( x ) = 2 / (3- 2 x ) on the interval- 3 / 2 < x < 3 / 2 (see Figure 1)! Indeed, if we cared to calculate many terms if the series, we would find that the difference on the interval- 3 / 2 < x < 3 / 2 between the partial sums and the limiting function becomes negligible. We can correctly say that the sequence converges to the limit on the given interval; however, since we know that the series values do not converge outside of this region, the convergent behaviour is not replicated outside of this region....
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- Spring '09
- Calculus, lim