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Conclude if the given series converges or diverges and justify the conclusion

by RoRi

Test the following series for convergence or divergence. Justify the decision.

    \[ \sum_{n=1}^{\infty} \frac{1}{1000n + 1}. \]

The series diverges. We use the limit comparison test, letting

    \[ a_n = \frac{1}{1000n + 1}, \qquad b_n = \frac{1}{n}. \]

Then we have

    \begin{align*} \lim_{n \to \infty} \frac{a_n}{b_n} &= \lim_{n \to \infty} \frac{\frac{1}{1000n + 1}}{\frac{1}{n}} \\[9pt] &= \frac{1}{1000}. \end{align*}

Thus, \sum a_n and \sum b_n both converge or both diverge, but \sum b_n diverges; hence, \sum a_n diverges as well.

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