Question

In Exercises \(53-68,\) determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n+10}{10 n+1} $$

Short answer

The series \(\sum_{n=1}^{\infty} \frac{n+10}{10n+1}\) diverges.

Step-by-step solution

Identify the Series Type

Identify the general term of the series, which in this case is \(a_n = \frac{n+10}{10n+1}\). The series involves a rational expression, which suggests that we could apply the comparison test.

Apply The Limit Comparison Test

Choose a second series that we can compare with our original series. A good choice is \(b_n = \frac{n}{n}=1\), as it is a simple series to compare with and we can determine its convergence easily. Apply the Limit Comparison Test

Calculate The Limit

Calculate the limit as n approaches infinity of \(\frac{a_n}{b_n}\), which in this case is \(\frac{\frac{n+10}{10n+1}}{1}\). By direct simplification, we have \(\lim_{n\to\infty} \frac{n+10}{10n+1} = \lim_{n \to \infty} \frac{n}{10n} = \frac{1}{10}\)

Interpret the result

Since the limit is a finite positive number, according to the limit comparison test, the original series has the same convergence behavior as the series we chose for comparison, which in this case is \(\sum_{n=1}^{\infty} 1\). This series is a harmonic series that diverges.