Final answer:
The sequence (x_n) = 1/(n+1) + 1/(n+2) + ... + 1/(2n) does not converge, as it is always greater than 1/2, suggesting divergence as n approaches infinity.
Step-by-step explanation:
The question asks whether the sequence (xn), defined by xn := 1/(n+1) + 1/(n+2) + … + 1/(2n) for n in the natural numbers N, converges or diverges. To analyze the convergence of this sequence, we compare it to a known convergent series. If we consider the harmonic series, we know it diverges; however, our sequence has fewer terms and each term is smaller than the corresponding term in the harmonic series for sufficiently large n. Nonetheless, since each term in our sequence is greater than 1/(2n), we can establish a comparison with the sum of n such terms. This simplifies to n/(2n) which is 1/2. As our sequence is always greater than 1/2, it suggests that xn does not approach 0 as n becomes large, pointing towards divergence.
The provided reference information about foci and binomial theorem expansions does not appear to be directly related to the question on sequence convergence and may be part of the typos or irrelevant details we were instructed to ignore.