Taylor Series Taylor's Theorem Additional exercises 12 Three Dimensions 1. The Coordinate System 2. Vectors 3. The Dot Product 4. The Cross Product 5. Lines and Planes 6. Other Coordinate Systems 13 Vector Functions 1. Space Curves 2. Calculus with vector functions 3. Arc length and curvature 4. Motion along a curve 14 Partial Differentiation 1. Functions of Several Variables 2. Limits and Continuity 3. Partial Differentiation 4.
The Chain Rule 5. Directional Derivatives 6. Higher order derivatives 7. Maxima and minima 8. Lagrange Multipliers 15 Multiple Integration 1. Volume and Average Height 2. Double Integrals in Cylindrical Coordinates 3. Moment and Center of Mass 4. Surface Area 5. Triple Integrals 6. Cylindrical and Spherical Coordinates 7. Change of Variables 16 Vector Calculus 1. Vector Fields 2. Line Integrals 3. The Fundamental Theorem of Line Integrals 4. Green's Theorem 5.
Divergence and Curl 6. Vector Functions for Surfaces 7. Surface Integrals 8. Stokes's Theorem 9. The Divergence Theorem 17 Differential Equations 1. First Order Differential Equations 2. First Order Homogeneous Linear Equations 3. From this we can get the radius of convergence and most of the interval of convergence with the possible exception of the endpoints.
With all that said, the best tests to use here are almost always the ratio or root test. The limit is then,. So, we have,. Notice that we now have the radius of convergence for this power series.
These are exactly the conditions required for the radius of convergence. All we need to do is determine if the power series will converge or diverge at the endpoints of this interval. The way to determine convergence at these points is to simply plug them into the original power series and see if the series converges or diverges using any test necessary.
So, in this case the power series will not converge for either endpoint. The interval of convergence is then,. The power series could converge at either both of the endpoints or only one of the endpoints. We need to be careful here in determining the interval of convergence. In other words, we need to factor a 4 out of the absolute value bars in order to get the correct radius of convergence.
For example, the geometric series converges for all x in the interval but diverges for all x outside that interval. We now summarize these three possibilities for a general power series. Consider the power series The series satisfies exactly one of the following properties:.
Suppose that the power series is centered at For a series centered at a value of a other than zero, the result follows by letting and considering the series We must first prove the following fact:. If there exists a real number such that converges, then the series converges absolutely for all x such that. Since converges, the n th term as Therefore, there exists an integer N such that for all Writing. Suppose that the set Then the series falls under case i. Suppose that the set S is the set of all real numbers.
Then the series falls under case ii. Suppose that and S is not the set of real numbers. Then there exists a real number such that the series does not converge. Thus, the series cannot converge for any x such that Therefore, the set S must be a bounded set, which means that it must have a smallest upper bound. This fact follows from the Least Upper Bound Property for the real numbers, which is beyond the scope of this text and is covered in real analysis courses.
Call that smallest upper bound R. Since the number Therefore, the series converges for all x such that and the series falls into case iii. If a series falls into case iii. Since the series diverges for all values x where the length of the interval is 2 R , and therefore, the radius of the interval is R. The value R is called the radius of convergence. For example, since the series converges for all values x in the interval and diverges for all values x such that the interval of convergence of this series is Since the length of the interval is 2, the radius of convergence is 1.
Consider the power series The set of real numbers x where the series converges is the interval of convergence. If there exists a real number such that the series converges for and diverges for then R is the radius of convergence.
If the series converges only at we say the radius of convergence is If the series converges for all real numbers x , we say the radius of convergence is Figure. To determine the interval of convergence for a power series, we typically apply the ratio test. In Figure , we show the three different possibilities illustrated in Figure. For each of the following series, find the interval and radius of convergence. Therefore, the series converges for all real numbers x. The interval of convergence is and the radius of convergence is Apply the ratio test.
For we see that Therefore, the series diverges for all Since the series is centered at it must converge there, so the series converges only for The interval of convergence is the single value and the radius of convergence is In order to apply the ratio test, consider The ratio if Since implies that the series converges absolutely if The ratio if Therefore, the series diverges if or The ratio test is inconclusive if The ratio if and only if or We need to test these values of x separately.
For the series is given by. Since this is the alternating harmonic series, it converges. Thus, the series converges at For the series is given by. This is the harmonic series, which is divergent. Therefore, the power series diverges at We conclude that the interval of convergence is and the radius of convergence is Find the interval and radius of convergence for the series. The interval of convergence is The radius of convergence is.
Polynomial functions are the easiest functions to analyze, since they only involve the basic arithmetic operations of addition, subtraction, multiplication, and division. If we can represent a complicated function by an infinite polynomial, we can use the polynomial representation to differentiate or integrate it.
In addition, we can use a truncated version of the polynomial expression to approximate values of the function. So, the question is, when can we represent a function by a power series? As a result, we are able to represent the function by the power series. We now show graphically how this series provides a representation for the function by comparing the graph of f with the graphs of several of the partial sums of this infinite series.
Sketch a graph of and the graphs of the corresponding partial sums for on the interval Comment on the approximation as N increases. From the graph in Figure you see that as N increases, becomes a better approximation for for x in the interval. Sketch a graph of and the corresponding partial sums for on the interval.
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