Let and consider

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let cn=narctan⁡ ⁣(1n)−1c_n = n\arctan\!\left(\tfrac{1}{n}\right) - 1 and consider ∑n=1∞cn\displaystyle\sum_{n=1}^{\infty} c_n. What does the limit of cnc_n imply about the series? (a) The series diverges because lim⁡n→∞cn=−1\lim_{n\to\infty} c_n = -1. (non-zero limit) (b) The test is inconclusive because lim⁡n→∞cn=0\lim_{n\to\infty} c_n = 0. (limit equals zero) (c) The series converges because lim⁡n→∞cn=0\lim_{n\to\infty} c_n = 0. (zero limit implies convergence) (d) The test is inconclusive because lim⁡n→∞cn\lim_{n\to\infty} c_n does not exist.

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