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Power Series

Power Series. A power series in x centered at 0 is a series of the form a 0 + a 1 x + a 2 x 2 + a 3 x 3 + … where a 0 , a 1 , a 2 , … are real numbers.

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Power Series

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  1. Power Series A power series in x centered at 0 is a series of the form a0 + a1x + a2x2 + a3x3 + … where a0 , a1,a2 , … are real numbers. The major characteristic of a power series is that it involves a variable, and its convergence depends on the value that x takes. Consequently, it is important to know the set of numbers that make the power series converge.

  2. General form of a power series is a0 + a1(x - c) + a2(x - c)2 + a3(x - c)3 + … where c is also a real number. We say that this is a power series in x centered at c (the reason will be obvious later). • Observations: • The above power series will definitely converge when x = c. • Any finite partial sum of the above power series is simply a polynomial in x.

  3. For any given power series a0 + a1(x - c) + a2(x - c)2 + a3(x - c)3 + … the set of numbers that make it converge is actually an interval on the real number line centered at c. More precisely, • Theorem • For any given power series of the above form, exactly one • of following is true. • It is convergent only for x = c. • It is convergent for all x. • There is a positive number R such thatit is convergent when |x - c| < R , andit is divergent when |x - c| > R.

  4. As a consequence of the previous theorem we have the following Definition The set {a : the power series converges when x is replaced by a} is called the interval of convergence of the power series. The number R in the previous theorem is called the radius ofconvergence for the given power series. In case (1), the radius of convergence is 0 and in case (2), the radius is .

  5. Computation for theradius of convergence Given a power series in its general form a0 + a1(x - c) + a2(x - c)2 + a3(x - c)3 + … the radius of convergence is provided that it exists or is equal to .

  6. Sum and Difference of power series Suppose that we have two functions defined by power series f(x) = a0 + a1(x - c) + a2(x - c)2 + a3(x - c)3 + … with radius of convergence R1 and g(x) = b0 + b1(x - c) + b2(x - c)2 + b3(x - c)3 + … with radius of convergence R2 , then and the radius of convergence is (i) min{R1, R2} if R1≠ R2 (ii) ≥ R1 if R1 = R2

  7. Product of power series Suppose that we have two functions defined by power series f(x) = a0 + a1(x - c) + a2(x - c)2 + a3(x - c)3 + … with radius of convergence R1 and g(x) = b0 + b1(x - c) + b2(x - c)2 + b3(x - c)3 + … with radius of convergence R2 , then f(x)g(x) = a0 b0 + [a1 b0 +a0 b1 ](x - c) + [a2 b0 + a1b1 + a0 b2](x - c)2 + [a3 b0 + a2 b1 + a1b2 + a0 b3](x - c)3 + … and the radius of convergence is min{R1 , R2} unless either f(x) = 0 or g(x) = 0.

  8. Reciprocal of a Power series Theorem: Suppose that we have a function defined by power series f(x) = a0 + a1(x - c) + a2(x - c)2 + a3(x - c)3 + … with radius of convergence R and a0 ≠ 0, then with positive radius of convergence ≤ R and further more

  9. Using Long Division to find the reciprocalof a power series Example: We know that and cos0 ≠ 0 , therefore we have a power series of secx around the point x = 0. This power series can be found by the long division

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  11. Differentiation of Power series If f(x) is represented by a power series a0 + a1(x - c) + a2(x - c)2 + a3(x - c)3 + … with radius of convergence R, then f(x) is continuous and differentiable in the interval (c - R, c + R), and f ´(x) = = a1 + 2a2(x - c) + 3a3(x - c)2 + 4a4(x - c)3 + … with the same radius of convergence (but the behavior at the endpoints may be different).

  12. Integration of Power series If f(x) is represented by a power series a0 + a1(x - c) + a2(x - c)2 + a3(x - c)3 + … with radius of convergence R, then f(x) is continuous and integrable in the interval (c - R, c + R), and with the same radius of convergence (but the behavior at the endpoints may be different).

  13. Representations offunctionsby power series The use of power series to represent a function was first discovered by long division, i.e. However, it was not understood that the above equality is valid only within the interval of convergence (-1, 1). Consequently, lots of absurd results were obtained such as

  14. Within the interval of convergence, we can produce many useful results from Replacing x by –x, we have Integrating both sides, we have

  15. But it is well know that Hence and by putting x = 0, we see that C = 0, And since the alternating series on the right is convergent at x = 1, we have (from a deeper result on next slide)

  16. Abel’s limit theorem for –r < x < r . Assume that we have If the series on the right is also convergent for x = r, then the limit exists and we have

  17. Replacing x by x2 in We have Integrating both sides, we have Actually C = 0 because tan-10 = 0 Replace x by 1, we have the interesting result

  18. Taylor Series of a Function Suppose that a given function f(x) does have a power series representation f(x) = a0 + a1(x - c) + a2(x - c)2 + a3(x - c)3 + … and we would like to find its coefficients an. We further have to assume that f(x) is infinitely differentiable at the point c, then it’s not hard to see that f(c) = a0 , f (c) = a1 , f (c) = 2a2 , f (3)(c) = 6a3 , … … f (n)(c) = n!an thus

  19. Taylor Series of a Function Definition: Suppose that f(x) is infinitely differentiable in a neighborhood of a point c, then the Taylor Series of f at the point c is • Remark: • The above series may not convergent for x c, • Even if the above series converges everywhere, it may not always converge to f(x). • Fortunately, most functions we will use are nice enough to have a Taylor series that really converges to f(x) in a non-trivial interval.

  20. Maclaurin Series of a Function Definition: Suppose that f(x) is infinitely differentiable in a neighborhood of a point c, then the Maclaurin Series of f is the Taylor series of f at x = 0, i.e.

  21. Maclaurin Series of a Function Famous examples: The radius of convergence for these series are all , and the equalities hold for all values of x.

  22. An Application of Power Series- the solutions to differential equations Given a differential equation we need to find a function f(x) that satisfies this equation.

  23. An Application of Power Series- the solutions to differential equations Given a differential equation we need to find a function f(x) that satisfies this equation.

  24. we need to find a function f(x) that satisfies this equation. Substituting these power series to the equation we have

  25. Substituting these power series to the equation we have = 0 Comparing coefficients we can conclude that which can be simplified to

  26. which can be further simplify to From the original condition, we know that therefore

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