(Solved): Which one is correct? If we are using the ratio test on a series \( \sum a_{ ...

Which one is correct?

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If we are using the ratio test on a series \( \sum a_{n} \), and we determine that \( \lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=0.8 \), what can we conclude? The convergent value of the entire series is \( \sum a_{n}=0.8 \). The series \( \sum a_{n} \) is similar enough to a geometric series with common ratio \( 0.8 \) that it has similar convergence behavior, and hence converges. The limit of the terms of the series is \( 0.8 \), which is not zero, meaning that the series diverges due to the Divergence Test.