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

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
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Solution

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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 \(\frac{8}{9} < 1\).

  1. Identify the terms of the series to compare.
  2. Compute the limit of the ratio of the given series term and the comparison series term as \(n\) approaches infinity.
  3. 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.

Step 2: Choose a Comparison Series

We choose the geometric series

\[ \sum_{n=1}^{\infty} \left(\frac{8}{9}\right)^{n} \]

which converges since \(\frac{8}{9} < 1\).

Step 3: Compute the Limit

We compute the limit

\[ L = \lim_{n \rightarrow \infty} \frac{\frac{8^{n}+1}{9^{n}+1}}{\left(\frac{8}{9}\right)^{n}}. \]

After evaluating this limit, we find that

\[ L = 1. \]

Step 4: Apply the Limit Comparison Test

Since \(L > 0\) and the comparison series converges, by the Limit Comparison Test, the original series

\[ \sum_{n=1}^{\infty} \frac{8^{n}+1}{9^{n}+1} \]

also converges.

Final Answer

The series converges, so the answer is

\(\boxed{\text{converges}}\).

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