8. Power Series
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8. Power Series A power series is a polynomial with infinitely many terms. Here is an example: Like a polynomial, a power series is a function of . That is, we can substitute in different values of to get different results. For example, . and . Though power series may seem complicated, they are actually not much more difficult to deal with than polynomials. For example, it is easy to take the derivative of a power series: and it is just as easy to take the integral: As you can see, a power series is not much more complicated than a polynomial. Taylor Series Consider again the power series . As we have seen, it is easy to compute the sum of this series for different values of . For example, . In fact, no matter what value of we plug in, the result is always a geometric series. The first term of this series is , and the common ratio is always . Indeed, the entire power series can be thought of as a geometric series with a common ratio of . This gives us a simple formula for the sum: This is our first example of a Taylor series—a power series that adds up to a known function. So which functions can be expressed as power series? The answer may surprise you: cos sin tan ln All of these functions—the exponential, the sine, the cosine, the logarithm, the inverse tangent—all of them are really just infinite polynomials! This result seems to good to be true. How is it possible that something like is an infinite polynomial? According to the table above, . Why would this be true? Well, the defining property of is that it is equal to its own derivative: . However, the Taylor series for is also equal to its own derivative: . Based on this observation alone, it seems reasonable that the two functions might be the same. Unfortunately, we are not yet in a position to fully explain the formulas above. The goal of the next two sets of notes is to learn how to express various functions as power series. By the time we are done, you will understand all five of these formulas. Geometric Power Series Recall the formula for the sum of a geometric series: As we have seen, this formula works perfectly well when and are functions of . For example, plugging in and gives the formula . There are many more series we can get this way. For example, using and gives , and using and gives, EXAMPLE 1 Find a formula for the sum of the following series: SOLUTION This is a geometric series with a common ratio of . The first term is , so EXAMPLE 2 Find a power series representation for each of the following functions: (a) (b) SOLUTION (a) This should be the sum of a geometric series with and : (b) This is the sum of a geometric series with and : Differentiation and Integration You can differentiate an integrate power series term-by-term, just as you would a polynomial: EXAMPLE 3 SOLUTION Find a power series representation for . Observe that: But: Therefore: We now come to our first major task: finding a power series for the natural logarithm. EXAMPLE 4 Find a Taylor series for ln . Observe that SOLUTION But ln . is the sum of a geometric power series: Integrating both sides gives: ln Plugging in reveals that . Therefore: ln The formula we just derived is our first really important result: ln We can find a power series for the inverse tangent using the same method. EXAMPLE 5 SOLUTION Find a Taylor series for tan . Recall that But tan . is the sum of a geometric series with and : . Integrating both sides gives: tan Plugging in reveals that , so: tan tan The result was: EXAMPLE 6 Express the integral: as the sum of an infinite series. Note that this integral would be very difficult to evaluate on its own. However, the integrand is the sum of a geometric series: SOLUTION Therefore: Substitution and Multiplication There are two more important tricks for working with power series. The first is substitution: EXAMPLE 7 SOLUTION Find a power series representation for tan . The power series for tan is: tan All we need to do is substitute in for : tan Though this method is very simple, it often comes off as confusing because of the two different 's. The idea here is that: tan for any . All we are doing is substituting in . You can think of many geometric series this way. For example, the series: can be obtained by substituting into the power series for EXAMPLE 8 SOLUTION . Find a power series representation for ln . We know that: ln Substituting in for yields: ln Power series can also be added, subtracted, and multiplied like polynomials. EXAMPLE 9 SOLUTION Find a power series representation for tan . We have: tan Summations for Power Series It is sometimes difficult to express a power series in summation notation. We give a few examples. EXAMPLE 10 Express the series: using summation notation. For this series, it seems easiest to have the first term be , the second term be , and so on: SOLUTION th term . The power of is increasing by each time, so it should be similar to . Indeed, it looks like the power of is , so: As you can see, the coefficient is always EXAMPLE 11 Express the series: tan using summation notation. SOLUTION This time we start with : th term The alternating and signs can be taken care of with a . It is important here that the even-numbered terms are positive (which is why we decided to start at ). If the odd numbered terms were positive, we would need a . The power of is an arithmetic sequence that increases by each time. In particular, the formula for the power is, so it should be similar to , so tan The following table shows the summation notation for each of our five primary series: cos sin tan ln Convergence of Power Series Consider a power series, say . Does this series converge? This is a question that we have been ignoring, but it is time to face it. Whether or not this power series converges depends on the value of . If is too large, then the series will diverge: . However, if is small enough, then the series will converge: In fact, since this particular series is geometric, it will converge whenever , and diverge whenever . Usually, the best way to determine convergence of a power series is to use the root test. EXAMPLE 12 For what values of does the following series converge? SOLUTION This is the series . Using the root test, lim . The series converges for , which happens when . Similarly, the series diverges when , which happens when . This gives us the following picture: DIVERGES ? CONVERGES ? DIVERGES All we need to determine is what happens for . (The root test gives for these two values of , which is inconclusive.) Here is what the series looks like for these values of : : : As you can see, the series diverges for both and . Therefore, the power series converges if and only if . For the series above, the root test determines that the series converges for and diverges for . This is always the sort of information that the root test provides. RADIUS OF CONVERGENCE Let be a power series. Then there exists a radius for which (a) The series converges for , and (b) The series converges for . is called the radius of convergence. Do not confuse the capital (the radius of convergence) with the lowercase (from the root test). They are completely different. stands for radius. In the last example, turned out to be , which resulted in a radius of . EXAMPLE 13 Find the radius of convergence for the series SOLUTION Using the root test: lim As you can see, the series converges if , and diverges if . Therefore, the radius of convergence is . EXAMPLE 14 Find the radius of convergence for the series SOLUTION Using the root test: lim As you can see, the series converges if , and diverges if . Therefore, the radius of convergence is . EXAMPLE 15 Find the radius of convergence for the series SOLUTION Using the root test: lim Since no matter what is, the series converges for any value of . Therefore, the radius of convergence is . EXERCISES 1–4 Find a power series representation for the function using the formula for the sum of a geometric series. 25. 1. 2. 26. 3. 4. 27. 5–16 Find a power series representation for the given function. 5. ln 6. tan 7. ln 8. ln 9. 10. sin 11. cos 12. 13. 14. tan 28. 29–40 Find the radius of convergence of the series. 29. ln tan 15. sin 16. 17–22 Express the integral as an infinite series. 35. 17. 37. 19. 1 21. 18. 20. ln 22. ln sin 39. 31. 33. 30. 32. 34. 36. 38. 40. ln ln 41. Suppose that the power series converges for 23–28 Express the given series using summation notation. 23. 24. , but diverges for . (a) Does the series converge when ? Explain. (b) Does the series converge when ? Explain.
