Consider the series

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

Problem

Consider the series ∑n=4∞5n2+1n3\displaystyle \sum_{n=4}^{\infty}\frac{5n^{2}+1}{n^{3}}. Which statement about its convergence is correct? (a) The series converges because the n3n^{3} in the denominator overpowers the numerator. (b) The information given is insufficient; another convergence test is required to draw a conclusion. (c) The series diverges because, for large nn, the terms behave like 1n\tfrac{1}{n}, the harmonic series. (d) The series converges by comparison with ∑1n3\displaystyle \sum\frac{1}{n^{3}}, a pp-series with p=3p=3.

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