The answer is 1/47 (D), but I am not sure how to do this problem at all.

Step-by-step explanation:
For an alternating series ∑a(n), the error bound of the sum of the first k terms is:
E < | a(k+1) |
Here, k = 10:
E < | a(11) |
E < | (-1)¹¹⁺¹ × 1 / (4(11) + 3) |
E < | 1/47 |
E < 1/47