1. The problem statement, all variables and given/known data Consider the following convergent series. Then complete parts a throw d below. sum[k=1,inf] 5/k^7 a. Find an upper bound for the remainder in terms of n 2. Relevant equations Estimating Series with Positive Terms Let f be a continuous, positive, decreasing function for x >= 1 and let a_k = f(k) for k = 1,2,3,.... Let S = sum[k=1,inf] a_k be a convergent series and let S_n = sum[k=1,n] a_k be the sum of the first n terms of the series. The remainder R_n = S - S_n satisfies R_n <= integral[n,inf] f(x)dx. Furthermore, the exact value of the series is bounded as follows: S_n + integral[n+1,inf] f(x)dx <= sum[k=1,inf] a_k <= S_n + integral[n,inf] f(X)dx 3. The attempt at a solution I'm unsure how to do this problem. I believe that I'm trying to evaluate S_n + integral[n,inf] f(X)dx I have no problem find the value of integral[n,inf] f(X)dx but am not sure how to find the value of S_n. I would now how to find the value of this if I was asked to find upper bound for the error for the first 50 terms, I could then find S_50 by just finding the sum which would be a finite number, but I am unsure how to find the upper bound in this case were I guess I'm trying to find the value of S_n in this case would be S_inf which I'm not sure how to do. Thank's for any help which you can provide me with.