Answer:
C, A, D, B
Step-by-step explanation:
(a) We need to compare the sum of this series to 3/4 and 31/36. So we just need to find the sum within 1/36.
1 / (n + 1 + 1)² ≤ 1/36
(n + 2)² ≥ 36
n + 2 ≥ 6
n ≥ 4
So we just need to find the partial sum of the terms from n=0 to n=4.
1 − 1/4 + 1/9 − 1/16 + 1/25 ≈ 0.839
We can use this to approximate the actual sum within 1/64.
(b) 1 − 1/7 + 1/49 − 1/343 + ...
We can rearrange this as the difference between two geometric series:
(1 + 1/49 + 1/2401 + ...) − (1/7 + 1/343 + 1/16807)
∑₀°° (1/49)ⁿ − ∑₀°° 1/7 (1/49)ⁿ
1 / (1 − 1/49) − (1/7) / (1 − 1/49)
(6/7) / (48/49)
7/8
So from smallest to largest, the answer is C, A, D, B.
