Remainder Estimate for the Integral Test

Question

The series ∑n=2 to ∞ for 1/(n(ln(n))^4) is convergent.

a) According to the Remainder Estimate for the Integral Test, the error in the approximation s ≈ s_n (where s is the value of the infinite sum and s_n is the n-th partial sum) is |s − s_n| ≤ _____. (find the value to fill in the blank)

b) Find the smallest value of n such that this upper bound is less than 0.07.

Answer 1

Part A: Find the Remainder Estimate

1. Identify the function:
2. We have
3. f (x) = 1
4. x(1n x)^4•
5. Find the integral: We need to compute the integral from n to ∞:
6. ∫^∞_n f (x) dx = ∫^∞_n f (x) = 1 dx ·
7. x(1n x)⁴
8. Use substitution:
9. Let u = 1n⁡, x then du = 1⁄x dx which means dx = e^u du. dx = du.
10. Change the limits:
11. When x = n, u = 1n⁡ n.
12. When x = ∞, u = ∞.
13. The integral becomes:
14. ∫^∞_{1n n} 1 du ·
15. u⁴
16. Evaluate the integral:
17. The integral ∫ 1 du = – 1⁄3u³ ·
18. u⁴
19. ^So,
20. ∫^∞_{1n n 1} du = [ – 1 ]^∞_{1n n = 0 – ( – 1} ) = 1
21. u⁴ 3u³ 3(1n n³ 3 (1n n)^{3 ·}
22. Remainder estimate:
23. The error is:
24. | s – s_n | ≤ 1
25. 3 (1n n)^{3 ·}
26. Part B: Find the smallest value of n
27. Set up inequality:
28. We want
29. 1 < 0.07.
30. 3 (1n n)³