$\text { Determine if the series } \sum _ { n = 1 } ^ { \infty } a _ { n } \text { defined by the formula converges or diverges. }$
-$$a _ { 1 } = 5 , a _ { n + 1 } = \frac { n } { n + 1 } a _ { n }$$
A) Converges
B) Diverges

Question

Quizplus · Accepted Answer

We can use the Ratio Test to determine the convergence of the series:
$$ \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n\to\infty} \left| \frac{n}{n+1} \right| = 1 $$
Since the limit is equal to 1, the Ratio Test is inconclusive and we need to use another test. Notice that for $n\geq 2$, we have
$$ a_n = \frac{n-1}{n} a_{n-1} < a_{n-1} $$
Therefore, the sequence $\{a_n\}_{n=1}^\infty$ is decreasing. Additionally, we have $a_n > 0$ for all $n\geq 1$. Thus, we can apply the Test for Divergence to conclude that the series $\sum_{n=1}^\infty a_n$ diverges.

Determine if the series ∑n=1∞an defined by the formula converges or diverges. \text { Determine if the series } \sum _ { n = 1 } ^ { \infty } a _ { n } \text { defined by the formula converges or diverges. } Determine if the series ∑n=1∞​an​ defined by the formula converges or diverges.

$\text { Determine if the series } \sum _ { n = 1 } ^ { \infty } a _ { n } \text { defined by the formula converges or diverges. }$