2022 Fermat Problem 20

A sequence of numbers t_1, t_2, t_3, \ldots has its terms defined by t_n = \dfrac{1}{n} - \dfrac{1}{n+2} for every integer n \ge 1. For example, t_4 = \dfrac{1}{4} - \dfrac{1}{6}. What is the largest positive integer k for which the sum of the first k terms (that is, t_1 + t_2 + \cdots + t_{k-1} + t_k) is less than 1.499?

\textbf{(A)}\ 2000\quad \textbf{(B)}\ 1999\quad \textbf{(C)}\ 2002\quad \textbf{(D)}\ 2001\quad \textbf{(E)}\ 1998

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.