Our starting point in this section is the geometric series. Does it make sense to think of a power series as a function of x. If a n and zare restricted to be real numbers, this is the usual power series that you are already familiar with. We consider a number of examples of power series and their radii of convergence. In this section well state the main theorem we need about the convergence of power series. Power series lecture notes a power series is a polynomial with infinitely many terms. Within its interval of convergence, the derivative of a power series is the sum of derivatives of individual terms. In this section, we are going to use power series to represent and then to approximate general functions. We look at a spike, a step function, and a rampand smoother functions too. Math 262 practice problems solutions power series and taylor. The series converges for jx aj r, where r0 is a value called the radius of convergence. I an equivalent expression for the power series is. If we assume that a solution of a di erential equation is written as a power series, then perhaps we can use a method reminiscent of undetermined coe cients. Examples of applications of the power series series.
That is, we can substitute in different values of to get different results. Use standard series, to expand the following functions in power series, as far as the terms shown. Whether or not this power series converges depends on the value of. If the power series only converges for x a then the radius of convergence is r 0 and the interval of convergence is x a. Here follows a collection of general examples ofpower series. We use the ratio test to obtain the radius of convergence r, of the power series and. Convergence of power series lecture notes consider a power series, say 0 b. The important technique of solving linear dierential equations with polynomial coecients by means of power series is postponed to the next book in this series,calculus 3c4. Power series and taylor series penn math university of. Likewise, if the power series converges for every x the radius of convergence is r \infty and interval of convergence is \infty 0.
Let us apply the ratio test and find the set of points in r on which the series converges. The reader is also referred to calculus 3b the important technique of solving linear differential equations with polynomial coefficients by means of power series is postponed to the next book in this series, calculus 3c4. A power series centered at z 0 2c is an expansion of the form x1 n0 a nz z 0n. These are examples of infinite series where each term contains a variable, x, raised to a positive integer power. An eigenvalue problem solved by the power series method 5 6 48 89 stand out from the crowd designed for graduates with less than one year of fulltime postgraduate work. This is a question that we have been ignoring, but it is time to face it. There is no reason to expect this power series to converge for all values of x. Power series solutions of differential equations examples. Here follows a collection of general examples of power series. Any power series fx p n n0 c nx an has one of three types of convergence. Example 3 for what values of does the following series converge. Solution of dierential equations by the power series method 2. As noted above, it is not always obvious that a series solution matches its analytic counterpart if it has one, so it is not expected that you will be able to relate every power series solution to a simple analytic form. Also state the range of values of x for which the power series converges.
It should no longer be necessary rigourously to use the adicmodel, described in calculus 1c and calculus 2c. For example, when x 0, this is a geometric series with common ratio. Technical details will be pushed to the appendix for the interested reader. For each of the following power series determine the interval and radius of convergence. Likewise, if the power series converges for every x the radius of convergence is r \infty and interval of convergence is \infty example. Click on exercise links for full worked solutions there are 10 exercises in total. For each of the following functions, find the taylor series about the indicated center and also determine the interval of convergence for the series. See how this is used to find the derivative of a power series.
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