Series Handout A. 1. Determine which of the following sums are geometric. If the sum is geometric, express the sum in closed form.

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1 Series Handout A. Determine which of the following sums are geometric. If the sum is geometric, exress the sum in closed form. 70 a) k= ( k ) b) 50 k= ( k )2 c) 60 k= ( k )k d) 60 k= (.0)k/3 2. Find the sum of each of the following. 00 a) k=0 ( 3 )k b) k=0 ( 3 )k c) 00 k=2 ( 3 )k d) k=2 ( 3 )k 3. Do the first few terms of a series affect whether or not the series converges? Do the first few terms of a convergent series affect its sum? 4. (a) For what values of x does the series k=0 xk converge? For these values of x, what function does the series converge to? (b) For what values of x does the series k=0 (x 4)k converge? For these values of x, what function does the series converge to? 5. What do you think? Some of our roblem sets will include What do you think? questions. You will receive full credit on your homework for a thoughtful answer to such a question - regardless of whether you have answered it correctly. Write your thoughts in en and leave some blank sace below your answer on your homework. These questions will be discussed in class and you can correct any mistaken ideas you had in encil below. The next two roblems ask you to make sense of the definition of a convergent series. Let a + a 2 + a a n + be an infinite series and let s n denote its nth artial sum: s n = a + a 2 + a a n. Definition: We say k= a n converges if lim n s n = S for a finite number S. We write k= a k = S and say k= a k converges to S. If lim n s n does not exist (or is not finite) then we say the series k= a n diverges. Suose you know that the infinite series a + a 2 + a a n + converges and that a k > 0 for k any ositive integer. For each of the following statements, determine whether the statement must be true, could ossibly be true, or must be false. (a) lim n a n = 0 (b) lim n s n = 0 (c) There exists a number M such that s n < M for all n. (This is equivalent to saying that the artial sums are bounded.) (d) k=5 a k converges 6. What do you think? Suose you know that lim n b n = 0. Does it necessarily follow that the infinite series k= b k converges? 7. Suose a n = f(n) where f is a function defined on (, ) and n is an integer. We know that if lim x f(x) = L, then lim n f(n) = L (so lim n a n = L) as well. Zenobia has the mistaken belief that lim n f(n) and lim x f(x) are always equal. Which of the following functions would be best to use to ersuade her to change her mind? (a) f(x) = sin x (b) f(x) = sin x x (c) f(x) = sin(πx)

2 8. What do you think? Write out the first few terms of the series k= 2 k + k. (This series is not a geometric series.) Now write out the first few terms of the geometric series k=. 2 k By comaring the terms of the two series, determine whether or not the former series converges. Exlain your reasoning in words carefully and clearly. Your answer will form the launching ad for the next class. 9. In class we studied a family of series known as series, series of the form n +. We know that x dx diverges for < and converges for >. We use the integral test to conclude that n + converges if > and diverges if <. Sometimes students incorrectly think that the series n= n is the dividing line for series of any sort, not just for series. This roblem is meant to disel that notion. (a) We know that n+2 < n for all ositive n. Nevertheless, show that the series n= n+2 diverges. (b) We know that < n2 +0 n for all ositive n. Nevertheless, show that the series n= diverges. n2 +0 (c) (otional - late addition) We know that In fact, n= Extra credit: Does n= n ln n < n for all ositive n. Nevertheless, show that the series n= n ln n diverges. n ln n diverges, but n= n(ln n) converges for any >. How can you show this? converge or does it diverge? n(ln n)(ln(ln n)) 0. If f is a function, then P n (x), the Taylor olynomial about x = 0 associated to f, is to be thought of as a good aroximation of f for ear zero. Let s consider the olynomial f(x) = x 3 4x 2 + 4x. (a) What do you think the best linear aroximation of this olynomial (aroximation of the form P (x) = a 0 + a x) ought to be at the oint b = 0? Why? What is the best quadratic aroximation of the olynomial f (aroximation of the form P 2 (x) = a 0 + a x + a 2 x 2 ) at the oint b = 0? What is the best cubic aroximation of the olynomial f (aroximation of the form P 3 (x) = a 0 + a x + a 2 x 2 + a 3 x 3 ) at the oint b = 0? Does this surrise you? Why or why not? (b) The number 2 is a critical oint of f. If we write P n (x) = a 0 + a (x 2) + a 2 (x 2) a n (x 2) n then a will be zero. Why? In addition, a 4, a 5,..., a n will all be zero. Why is that? (c) If b is any real number and we write P n (x) = a 0 + a (x b) + a 2 (x b) a n (x b) n then what can we say about a 4, a 5...? Why?. Let f(x) be a function defined on the domain (, ) with exactly one zero, which occurs at x = 3. The first and second derivatives of f exist everywhere on its domain. We know that f is increasing on (, ], decreasing on [, ), concave down on (, 2), and concave u on (2, ). Below are second degree Taylor olynomials for f centered at 0,, 2, 3, and 4 resectively. information to determine the sign (ositive, negative, or zero) or each of the coefficients. (a) a 0 + a x + a 2 x 2 (b) b 0 + b (x ) + b 2 (x ) 2 (c) c 0 + c (x 2) + c 2 (x 2) 2 (d) d 0 + d (x 3) + d 2 (x 3) 2 (e) e 0 + e (x + 4) + e 2 (x + 4) 2 Use the 2

3 2. By taking derivatives, show that the Taylor series for the function f(x) = x at x = 0 is the geometric series k=0 xk. (This illustrates the fact that if a function has a ower series exansion about x = a then that ower series exansion is its Taylor series about x = a.) 3. We know that the Taylor series generated by e x is to e x for all x. In other words, e x =. In fact, it can be shown that this series converges. By evaluating both sides of this equation at x = 0. you can obtain an exression for e 0.. How many terms of the series are needed to aroximate e 0. with error less than 0 8? 4. If sin x = ( ) k x2k+ for all x, then how many non-zero terms of the series exansion for sin(0.2) must (2k)! k=0 be used in order to aroximate sin(0.2) with error less than 0 6? Will this aroximation be too large, or too small? 5. The Alternating Series Test says that if an infinite series is (i) alternating, (ii) the magnitude of the terms is decreasing, and (iii) the magnitude of the terms tends to zero, then the series converges. This is a test for convergence only; don t try use it to show divergence. If the terms are not tending to zero, then the series diverges by the Nth term test for Divergence, not by the AST. If either one of the other two conditions is not satisfied then the test is inconclusive - meaning the series might converge or it might diverge. Below (for your reading leasure) we resent an alternating series whose terms are tending towards zero but not decreasing in magnitude. Conditions (i) and (iii) are satisfied by the series but n n + ( ) ( ) ( ) ( n ) + n can be written n + which is the harmonic series. The harmonic series diverges, so the series dislayed diverges. (It is also ossible to construct a convergent alternating series where the terms tend to zero but are not decreasing. You could concoct one of these by taking an absolutely convergent series whose terms are decreasing, making it alternating, and then changing the order of the terms so that it is no longer decreasing.) In the roblems below, determine whether or not the series converges. (The alternating series test cannot be invoked: why not?) (a) (b) ( ) k 2k2 0k 0k 2 + 5k k= ( ) k sin k k 3 k= 6. We know that the Taylor series generated by e x about x = 0 is this series converges for all x. 7. We know that the Taylor series generated by cos x about x = 0 is show that this series converges for all x.. Use the Ratio Test to show that ( ) n x2n. Use the Ratio Test to (2n)! 3

4 (x 3) n 8. The ower series 5 n is geometric. Show that the information given by the Ratio Test about convergence agrees with what we know about the convergence of a geometric series. 9. What exactly do we mean when we write k= a k = 4? Your answer can be brief, but must be recise and accurate. You will get full credit only if you use words correctly. 20. Suose that a ower series of the form k=0 c k(x 2) k has a radius of convergence of 7. What are the ossibilities for the interval of convergence of the series? 2. Suose the interval of convergence of the ower series c n(x a) n is ( 5, 7]. (a) What is a? (b) Does the series converge for x = 6.5? For x = 6.5? 22. The interval of convergence of the Taylor series for ln( + u) about x = 0 is u (, ]. On this interval the series converges to ln( + u). (a) Using any method you like, find the Maclaurin series for ln( + u). (b) By setting u = x in art (a), find a ower series exansion for ln x centered at x =. (c) Find the Taylor series for ln x at x = by taking derivatives. Make sure your answers to arts (b) and (c) agree. (They ought to because if a function has a ower series exansion in (x ) then that exansion will be the Taylor series about x =.) 23. In this roblem you ll comare the error analysis arrived at using the alternating series error estimate with that gotten using the Taylor Remainder. Let f(x) = x /3. (a) Find the third order Taylor Polynomial, T 3 (x), for f at x = 27. (b) Use T 3 (x) found in art (a) to aroximate 28 /3. (c) Find an uer bound for the error in this aroximation by using the alternating series error estimate. (d) Now find an uer bound for the error in this aroximation by using the Taylor Remainder (i.e., Taylor s Inequality). 24. Multilying Series: (a) Find the third degree Taylor olynomial (centered at 0) for f(x) = sin x cos x by multilying Taylor series for sin x and cos x and stoing when you know that all remaining terms are of degree 4 or higher. Check your answer by using the identity sin 2x = 2 sin x cos x. (b) In your last homework set you found the series for + x. (It was 8.8 #: the answer is in the back of the book.) Multily this series by itself to find the second degree Taylor olynomial for + x + x. Without comuting, what do you exect the coefficient of the x 3 term to be? 25. Amanda asked her friend Charlie for some hel when studying for a math test on series. Charlie had quick and easy methods, but he sometimes said things that are incorrect. Your job in this roblem is to ay very careful attention to Amanda and Charlie s conversation and correct Charlie wherever necessary. (a) Amanda asked Charlie how you can tell if the series, + ( ) + ( ) 2 + ( ) 3 + ( ) converges or diverges. Charlie answered, That s easy. This is just a geometric series with a = and r =. So you just lug into the formula, a r = ( ) = 2. So, it converges to one half. Do you agree with Charlie s statement? Exlain why or why not. 4

5 (b) Amanda told Charlie that she was having a lot of trouble with geometric series. She asked Charlie how you could find out whether an exression like: converged or not, and if it did, what it converged to. Charlie answered, Okay, these are two ste roblems. First, that s a geometric series with a = 3, r = 7 2. So, to get the total, you just lug into the formula: a( r n+ ) = 3( 492 ). r 49 Charlie told Amanda that you can work that out on a calculator. (Is Charlie s closed form correct? Exlain below.) You can tell the series doesn t converge Charlie continued, because the r is 49. That s greater than one, so the series doesn t converge. Is Charlie s statement about convergence accurate? Why or why not? (c) The last question that Amanda asked was about a general infinite series like k= a k. She said that she remembered hearing something about looking at the value of a k when k gets really, really big, and that can tell you something about whether the series k= a k converges or not. Charlie resonded, Oh yeah, this makes convergence and divergence really easy. If you look at what the formula for a k is, and if that formula gets closer and closer to zero then the series converges. Otherwise, the series diverges. Do you think Charlie is gave Amanda very solid advice? What advice would you have given her? 5