Consider the infinite series

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

Problem

Consider the infinite series ∑k=5∞1k−4\displaystyle \sum_{k=5}^{\infty}\frac{1}{k-4}. Which statement about its convergence is correct? (a) It diverges since its terms behave like 1k\dfrac{1}{k}, matching the harmonic series. (b) It converges because the index shift decreases each term enough to form a convergent series. (c) It converges as a pp\!-series with p>1p>1 after the index shift. (d) It converges by comparison with the convergent series ∑k=5∞1k2\displaystyle \sum_{k=5}^{\infty}\frac{1}{k^{2}}."

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