38510.3 Convergence Tests All Series 23 If Za, does not converge show that Zla,l does not 34 Verify the Schwarz inequality (C a, bJ2 <(C aZ)(Z bi) if converge. a, =(4)" and b, =(4)". a2 24 Find conditions which guarantee that a, +a, -a3+ 35 Under what condition does ?(a,+, -an) converge and a, +a5-a, + -.-will converge (negative term follows two what is its sum? positive terms). 36 For a conditionally convergent series, explain how the 25 If the terms of In 2 = 1-4 +3-f + --.are rearranged into terms could be rearranged so that the sum is + co. All terms 1-3-6 +4 - - & + --.,show that this series now adds to must eventually be included, even negative terms. 4 In 2. (Combine each positive term with the following nega- tive term.) 37 Describe the terms in the product (1 +4 +f + .--)(I+4 + 26 Show that the series 1 +4 -+4+4-+ converges 4 + ---)and find their sum. to 4 In 2. 38 True or false: 27 What is the sum of 1 +*-$+*-f +4-&+ -.a? (a) Every alternating series converges. 128 Combine 1+--. +--lnn+y and 1-$+~-.-+ln2 (b) Za, converges conditionally if Z la,l diverges. n (c) A convergent series with positive terms is absolutely to prove 1+*+4-3-$-&+ =ln2. convergent. 29 (a) Prove that this alternating series converges: (d) If Can and Cb, both converge, so does C(a, +b,). 39 Every number x between 0 and 2 equals 1 +4 +4+ ..-with suitable terms deleted. Why? (b) Show that its sum is Euler's constant y. 40 Every numbers between -1 and 1 equals +f f$ f$ f--. with a suitable choice of signs. (Add 1 =4+f +4 + --.to get 30 Prove that this series converges. Its sum is 42. Problem 39.) Which signs give s = -1 and s =0 and s =i? 41 Show that no choice of signs will make +4+4$&+ .--equal to zero. 131 The cosine of 8 = 1 radian is 1 --+-.-1 .--.Compute 42 The sums in Problem 41 form a Cantor set centered at 2! 4! cos 1 to five correct decimals (how many terms?). zero. What is the smallest positive number in the set? Choose signs to show that 4is in the set. It3 715 ma..32 The sine of 8 =7~ radians is n --3! +-5! -Compute "43 Show that the tangent of 0 =q(n -1) is sin 1/(1 -cos 1). sin 7~ to eight correct decimals (how many terms?). This is the imaginary part of s = -ln(1 -ei). From s =Z ein/n deduce the remarkable sum C (sin n)/n =q(7~-1).33 If Xai and Zbi are convergent show that Za,b, is abso- lutely convergent. 44 Suppose Can converges and 1x1 < 1. Show that Ca,xn Hint: (a fb)2 20 yields 2)abJ <a2+b2. converges absolutely. 10.4 The Taylor Series for ex,sin x, and cos x -This section goes back from numbers to functions. Instead of Xu, = s it deals with Xanxn=f(x). The sum is afunction of x. The geometric series has all a, = 1 (including a,, the constant term) and its sum is f(x) = 1/(1- x). The derivatives of 1 +x +x2 + --. match the derivatives off. Now we choose the an differently, to match a different function. The new function is ex. All its derivatives are ex. At x =0, this function and its derivatives equal 1. To match these l's, we move factorials into the denominators.10 Infinite Series Term by term the series is xn/n!has the correct nth derivative (= 1). From the derivatives at x = 0, we have built back the function! At x = 1 the right side is 1 + 1 + 4+ & + .-.and the left side is e = 2.71828 .... At x = -1 the series gives 1-1+ f -4+ -, which is e-'. The same term-by-term idea works for differential equations, as follows. EXAMPLE 1 Solve dyldx = -y starting from y = 1 at x = 0. Solution The zeroth derivative at x = 0 is thefunction itseg y = 1. Then the equation y' = -y gives y' = -1 and y" = -y' = + 1. The alternating derivatives 1, -1, 1, -1, ... are matched by the alternating series for e-": y = 1 -x + tx2 -ix3 + ... --e-X (the correct solution to y' = -y). EXAMPLE 2 Solve d'y/dx2 = -y starting from y = 1 and y' = 0 (the answer is cos x). Solution The equation gives y" = -1(again at x = 0). The derivativeof the equation gives y'" = -y1= 0. Then = -y" = + 1. The even derivatives are alternately + 1 and -1, the odd derivatives are zero. This is matched by a series of even powers, which constructs cos x: 1 1 1 y = 1 --X2 + --X6 + ... = cos X.2! 4! 6! The first terms 1 -$x2 came earlier in the book. Now we have the whole alternating series. It converges absolutely for all x, by comparison with the series for ex (odd powers are dropped). The partial sums in Figure 10.4reach further and further before they lose touch with cos x. Fig. 10.4 The partial sums 1 -x2/2 + x4/24 ---.of the cosine series. If we wanted plus signs instead of plus-minus, we could average ex and e-". The differential equation for cosh x is d2y/dx2= + y, to give plus signs: 1 1 1 1-(ex + e-") = 1 + -x2 + -x4 + -x6 + (which is cosh x).2 2! 4! 6! WOR SERIES The idea of matching derivatives by powers is becoming central to this chapter. The derivatives are given at a basepoint (say x = 0). They are numbersf(O),f '(O), .... The derivative f@)(O)will be the nth derivative of anxn,if we choose a, to be f(")(O)/n!0.4A The Taylor Series for eX, sin ;x and cos xThen the series I anx" has the same derivatives at the basepoint as f(x):10K The Taylor series that matches f(x) and all its derivatives at x = 0 is1 1 , 2....... 1, + + f(0) + f '(0)x + f"02 2 + '(0)x + = fn)(0)...6 n=O n!x.The first terms give the linear and quadratic approximations that we know well. Thex3 term was mentioned earlier (but not used). Now we have all the terms-an "infiniteapproximation" that is intended to equalf(x).Two things are needed. First, the series must converge. Second, the function mustdo what the series predicts, away from x = 0. Those are true for ex and cos x andsin x; the series equals the function. We proceed on that basis.The Taylor series with special basepoint x = 0 is also called the "Maclaurin series."EXAMPLE 3 Find the Taylor series for f(x) = sin x around x = 0.Solution The numbers f(")(0) are the values of f= sin x, f' = cos x, f" = -sin x,...at x = 0. Those values are 0, 1, 0, -1, 0, 1, .... All even derivatives are zero. To findthe coefficients in the Taylor series, divide by the factorials:sin x= x- ix3 ± + X5 -(2)EXAMPLE 4 Find the Taylor series forf(x) = …
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