Does the Series cos(n)/n^(3/4) Converge or Diverge?Date: 11/10/2009 at 21:30:44 From: Collin Subject: Is the series cos(n)/n^(3/4) convergent or divergent Is the series cos(n)/n^(3/4) convergent or divergent? This question is quite a doozy. I have tried the interval test and it was inconclusive. I tried the strong comparison test but could not think of something to compare it to. The limit comparison test gets too difficult as well. Date: 11/12/2009 at 22:11:01 From: Doctor Jordan Subject: Re: Is the series cos(n)/n^(3/4) convergent or divergent Hi Collin, The series cos(n)/n^(3/4) does, indeed, converge. However, I will note that this is similar to the question of whether the series cos (pi*n)/n^(3/4) is convergent. That question is #15 in section 12.5 of James Stewart's "Calculus," 6th ed. For the latter series, we note that cos(pi*n) = (-1)^n, then find that the series is alternating; and since the terms decrease to 0, conclude that that series converges. It is rather harder to show that cos(n)/n^(3/4) is convergent. Several times in the argument, we will have to bound expressions; for instance, 1 - cos(N + 1) <= 2. I will explain how to show this; and if you want more details, please write back. First we have to show that sum(cos(n), n = 0, ..., N) does not get too big. This makes sense because cos(n) is bouncing around being negative and positive, so a sum of many cos(n) will probably not be that large, whether positive or negative. It's not obvious at this point why we want to show that the sum of cos(n) from n = 0 to N does not get too big, but it will be useful later on in the proof. We use the fact that e^(ix) = cos(x) + i*sin(x). Then we use the finite geometric series: sum(x^n, n = 0, ..., N) = (1 - x^(N + 1))/(1 - x). Here, we put x = e^i. The real part of e^(i*n) is cos(n). We take the real part of the above fraction, which will thus be equal to sum(cos(n), n = 0, ..., N). After a few lines of work, we find that the real part of the fraction is (1 - cos(N + 1))*(1 - cos(1)) + sin(1)*sin(N + 1) ------------------------------------------------- (1 - cos(1))^2 + (sin(1))^2 So this is an expression for sum(cos(n), n = 0, ..., N). Let a_n = n^(-3/4) b_n = cos(n) B_N = sum(b_n, n = 1, ..., N) S_N = sum(a_n*b_n, n = 1, ..., N) The complicated fraction with cos and sin that we found above is equal to B_N + cos(1), since B_N is summing from n = 1 to N instead of from n = 0 to N. Therefore, (1 - cos(N + 1))*(1 - cos(1)) + sin(1)*sin(N + 1) B_N = -1 + ------------------------------------------------- (1 - cos(1))^2 + (sin(1))^2 Let || denote absolute value. We have to show that|S_N| has a limit as N -> infinity. Now, to upper bound a fraction, we have to find an upper bound for the numerator and a lower bound for the denominator. Since |cos(x)| <= 1 and |sin(x)| <= 1 for all x, using the triangle inequality we get the following: 2*2 + 1*1 |B_N| <= 1 + ----------- (sin(1))^2 Now, sin(1) > 4/5. So 5 |B_N| < 1 + ----- 16/25 = 1 + 125/16 = 141/16 < 9. The formula for summation by parts is S_N = a_N*b_N - sum(B_n*(a_(n + 1) - a_n), n = 1, ..., N - 1). Thus S_N <= a_N*b_N - 9*sum(a_(n + 1) - a_n, n = 1, ..., N - 1). But all the terms in the sum cancel except for the first and last, leaving S_N <= a_N*b_N - 9*(a_N - a_1). In other words, S_N <= N^(-3/4)*cos(N) - 9*N^(-3/4) + 9. As N -> infinity, the right hand side approaches 9. This shows that | S_N| converges as N -> infinity. That is, the series cos(n)/n^(3/4) converges. It doesn't show that its value is 9 -- only that its value is less than or equal to 9. This was not an easy problem at all, so if you have any questions please feel free to write back. - Doctor Jordan, The Math Forum http://mathforum.org/dr.math/ |
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