Power Series

Power Series - Power Series Weve spent quite a bit of time...

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Power Series We’ve spent quite a bit of time talking about series now and with only a couple of exceptions we’ve spent most of that time talking about how to determine if a series will converge or not. It’s now time to start looking at some specific kinds of series and we’ll eventually reach the point where we can talk about a couple of application of series. In this section we are going to start talking about power series. A power series about a , or just power series , is any series that can be written in the form, where a and c n are numbers. The c n ’s are often called the coefficients of the series. The first thing to notice about a power series is that it is a function of x . That is different from any other kind of series that we’ve looked at to this point. In all the prior sections we’ve only allowed numbers in the series and now we are allowing variables to be in the series as well. This will not change how things work however. Everything that we know about series still holds. In the discussion of power series convergence is still a major question that we’ll be dealing with. The difference is that the convergence of the series will now depend upon the value of x that we put into the series. A power series may converge for some values of x and not for other values of x . Before we get too far into power series there is some terminology that we need to get out of the way. First, as we will see in our examples, we will be able to show that there is a number R so that the power series will converge for, and will diverge for . This number is called the radius of convergence for the series. Note that the series may or may not converge if . What happens at these points will not change the radius of convergence. Secondly, the interval of all x ’s, including the end points if need be, for which the power series converges is called the interval of convergence of the series.
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These two concepts are fairly closely tied together. If we know that the radius of convergence of a power series is R then we have the following. The interval of convergence must then contain the interval since we know that the power series will converge for these values. We also know that the interval of convergence can’t contain x ’s in the ranges and since we know the power series diverges for these value of x . Therefore, to completely identify the interval of convergence all that we have to do is determine if the power series will converge for or . If the power series converges for one or both of these values then we’ll need to include
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Power Series - Power Series Weve spent quite a bit of time...

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