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1、1 CHAPTER:POWER SERIES Contents 0Introduction 1Maclaurins Theorem 2Standard expansions 0 Introduction A few minutes play with a scientific calculator will show that for most values of the variable x, the values of the functions sin x, cos x, ex, ln x, and so on, are not rational numbers or recogniza
2、ble irrational numbers expressible as roots. If the internal logic of a calculator or computer is limited to the basic arithmetic operations of addition, subtraction, multiplication, and division, how does it calculate the value of these and other such functions? Power series is a way of representin
3、g these functions by polynomial series which will allow us to find their respective approximate values to any desired degree of accuracy. Definition: A power series is a series of the form a0 + a1x + a2x2 +a3x3 + . . . + anxn + . . . to , where the coefficients a0, a1, a2, a3, . . . , are constants.
4、 If Sn = a0 + a1x + a2x2 +a3x3 + . . . + anxn , and if Sn S as n then the power series is said to be convergent and have a sum S. 1 Maclaurins Theorem Under certain conditions, a function f(x) can be expressed as an infinite series of ascending powers of x. Assuming that such a series exists, we may
5、 write f(x) = a0 + a1x + a2x2 +a3x3 + . . . + arxr + . . . where ai are constants. Differentiating with respect to x, we have, f(x)= a1 + 2a2x + 3a3x2 + 4a4x3 + . . . + rarxr-1 + . . . f(x)= 2a2 + 3.2a3x + 4.3a4x2 + . . . .+ r(r-1)arxr-2 + . . . f(x)= 3.2a3 + 4.3.2a4x+ . . . .+ r(r-1)(r-2)arxr-3 + .
6、 . . . f r(x)= r(r-1)(r-2) . . .3.2.ar + . . . Substituting x = 0, we find that f (0) = a0a0 = f (0) f (0) = a1a1 = f (0) f (0) = 2a2a2 = f (0) 2! f (0) = 3.2a3a3 = f (0) 3! f r(0) = r(r-1)(r-2) . . . 3.2arar = f r(0) r! Substituting these expressions for ai back into the original f(x), we get f(x)
7、= f(0) + f (0)x + x2 + x3 + . . . + xr + . . . f (0) 2! f (0) 3! f (r) (0) r! This result is called Maclaurins Theorem and the series obtained is known as the Maclaurins Series for f(x). It is possible to find a Maclaurins series for any function f(x) whose derivatives f (0), f (0), f (0), . . . can
8、 be determined. The series must converge to the sum f(x) in order to be useful. Hence, for many functions, Maclaurins Theorem holds only within a restricted range of values of x. 2 Example 1.1 Use Maclaurins theorem to obtain the expansion for f(x) = (1+x)n where n is a real number. 1 + nx + x2 + .
9、for - 1 1, ln = 2 ( + + + . . . ). n + 1 n - 1 1 n 1 3n3 1 5n5 Hence, calculate the value of ln 2 correct to 4 decimal places. 0.6930 Solution: 6 Example 2.4 Given that y = e cos x, show that = y cos x sin x. By further differentiation, or otherwise, show that the d2y dx2 dy dx first three non-zero
10、terms in the Maclaurin expansion of e cos x are e ex2 + ex4. 1 2 1 6 Solution: Example 2.5 The variables x and y are related by = 2xy 1, and y = 1 at x = 0. Show that, at x = 0, = 4 and dy dx d3y dx3 = 12. Find Maclaurins series for y up to and including the term in x4, and hence find an approximati
11、on to d4y dx4 the value of y at x = 0.1, giving your answer to 3 decimal places. y = 1 x + x2 x3 + x4 + . , y 0.909 2 3 1 2 Solution: 7 Example 2.6 Given that y = exsin x, express in the form kex sin (x + ), stating clearly suitable values of k and . Express dy dx the next 2 derivatives in similar f
12、orm. With the aid of Maclaurins theorem, or otherwise, express y as a series of ascending powers of x as far as the term in x3. Use this series to find an estimate of the value of e1/2 sin . k = , = ; y” = 2 ex sin (x + ); y = 2 ex sin (x + );y =x + x2 + ; 1 22 4 22 3 4 x3 3 19 24 Solution 8 Example
13、 2.7 (J84/1/21) Given that y = e , show that 4(1 + x) + 2 = y. By further differentiation of this result, or otherwise, 1 + x d2y dx2 dy dx show that the series expansion for y in ascending powers of x up to the term in x4 is y = e (1 + x + x3 + kx4 ) where k is a constant to be determined. k = 1 2
14、1 48 5 384 Solution 9 SUMMARY (Power Series) Maclaurins Theorem f(x) = f(0) + f (0)x + x2 + x3 + . . . + xr + . . . f (0) 2! f (0) 3! f (r) (0) r! Standard Expansions (1 + x)n= 1 + nx + x2 + . for - 1 x 1 n(n - 1) 2! (1 + x)-1= 1 - x + x2 - x3 + . + ( - 1)r xr + .for - 1 x 1 (1 - x)-1= 1 + x + x2 +
15、x3 + . + xr + .for - 1 x 1 ex= 1 + x + + + . + + .for all values of x x2 2! x3 3! xn n! ln (1 + x) = x - + - + .+ (- 1)r + 1 + . for - 1 x 1 x2 2 x3 3 x4 4 xr r sin x= x - + - + . + (- 1)r + .for all values of x in radians x3 3! x5 5! x7 7! x2r + 1 (2r + 1)! cos x= 1 - + - + . + (- 1)r + .for all values of x in radians x2 2! x4 4! x6 6! x2r (2r)! =
