Questions: Consider the following series.
sum from n=1 to infinity of (8^n+1)/(9^n+1)
Use the Limit Comparison Test to complete the limit.
limit as n approaches infinity of ((8^n+1)/(9^n+1))/□=L>0
Determine the convergence or divergence of the series.
converges
diverges
Transcript text: Consider the following series.
\[
\sum_{n=1}^{\infty} \frac{8^{n}+1}{9^{n}+1}
\]
Use the Limit Comparison Test to complete the limit.
\[
\lim _{n \rightarrow \infty} \frac{\frac{8^{n}+1}{9^{n}+1}}{\square}=L>0
\]
Determine the convergence or divergence of the series.
converges
diverges
Solution
Solution Steps
To determine the convergence or divergence of the given series using the Limit Comparison Test, we need to compare it with a known series. We can compare it with the geometric series \(\sum \left(\frac{8}{9}\right)^n\), which is known to converge because \(\frac{8}{9} < 1\).
Identify the terms of the series to compare.
Compute the limit of the ratio of the given series term and the comparison series term as \(n\) approaches infinity.
Use the result of the limit to determine the convergence or divergence of the series.
Step 1: Identify the Series
We are given the series
\[
\sum_{n=1}^{\infty} \frac{8^{n}+1}{9^{n}+1}
\]
and we will use the Limit Comparison Test to analyze its convergence.