We have seen sequences and series of constants. If the terms of a sequence being summed are power functions, then we have a **power series**, defined by
Note that most textbooks start with *n* = 0 instead of starting at 1, because it makes the exponents and *n* the same (if we started at 1, then the exponents would be *n* - 1). Also note that the constant *c* is called the center of the power series.

A power series is a function of *x* and will always converge for *x* = *c*, because all the terms except *a*_{0} become zero. So the question we want to ask about power series convergence is whether it converges for other values of *x* besides *c*. We can use the ratio test to find out the absolute convergence of the power series by examining the limit, as *n* approached infinity, of the absolute value of two successive terms of the sequence. If the limit does not exist, then the series only converges for *x* = *c*. If the limit is zero, then the series converges for all *x*. If there is a limit and it is of the form
then the series converges for a limited interval about *c*. In this case, *R* is called the **radius of convergence** and the series converges when | *x* - *c* | <* R*. Since the ratio test doesn't tell us whether the series converges when* |**x*| = *R*, we have to test those two cases separately. The resulting set of values for *x* where the series converges is called the **interval of convergence**.

See About the calculus applets for operating instructions. |

The applet shows the power series
Note that the graph only shows *P _{nmax}* (where

Select the second example from the drop down menu, showing
(again, *c* = 0 initially and *nmax *= 10 for the graph). Using the same ratio test as above, we can find that the limit of the ratio of successive terms is
so the radius of convergence is 3 in this case. Since this is a geometric series with common ration *x*/3, the series will converge when | *x* | < 3. You can see from the graph, by moving the *x* slider, that this roughly corresponds to the part of the graph where the black point doesn't head off to infinity (it doesn't correspond exactly, since we are only graphing the sum of 10 terms of the series instead of all the terms).

Select the third example from the drop down menu, showing
Note that in this case we used more complicated exponents to deal with the fact that some of the terms are missing. If we had written it in a form where the exponent on *x* was *n*, some of the terms would be zero and we would not be able to use the ratio test. Using this test on the series as written gives us:
Since the limit is zero, this series converges for all *x*. In fact, we will see later that it is equal to cos *x*. If you happen to zoom out, you will notice that the graph does not converge for all *x*. This is because we are only graphing 10 terms. If you move the *nmax* slider, you will notice that the graph converges on a smaller interval. Similarly, if you make *nmax* bigger, it will converge on a larger interval. If you could make *nmax *equal to infinity, it would converge for all *x*.

Select the fourth example, showing
Using the ratio test we get:
Since this limit does not not exist, the series only converges for *x* = 0 (which is *c* in this example). If you set *nmax* = 100, you will see more clearly on the graph that the interval of convergence is very small (warning: don't set *nmax* much bigger than this, as the software will take a long time to compute the series).

- Sequences
- Series
- Integral Test
- Comparison Test
- Limit Comparison Test
- Ratio Test
- Alternating Series and Absolute Convergence
- Power Series & Interval of Convergence
- Taylor Series & Polynomials
- Lagrange Remainder

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