Math  /  Calculus

QuestionUse the Ratio Test to determine whether the series is convergent or divergent. n=1nπn(5)n1\sum_{n=1}^{\infty} \frac{n \pi^{n}}{(-5)^{n-1}}
Identify ana_{n}. \square
Evaluate the following limit. limnan+1an\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| \square
Since limnan+1an\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| ? 1, ---Select--- \square

Studdy Solution
Apply the Ratio Test. The Ratio Test states that a series an\sum a_n converges if:
limnan+1an<1\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1
Here, we found:
π50.628\frac{\pi}{5} \approx 0.628
Since π5<1\frac{\pi}{5} < 1, the series converges.
The series is convergent.

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