(a(n+1) / a(n)). In our case, the sequence a(n) is defined as a(n) = (n + 2) / 5^(n + 3).
So, we begin by finding a(n+1), which is: a(n+1) = ((n + 1) + 2) / 5^((n + 1) + 3)
This simplifies to: a(n+1) = (n + 3) / 5^(n + 4)
Now, we need to compute the ratio a(n+1) / a(n) which is:
[(n + 3) / 5^(n + 4)] / [(n + 2) / 5^(n + 3)]
To simplify this, let's multiply the numerator and denominator by 5^(n + 4), which results in:
(n + 3) / (n + 2) * (5^(n + 3) / 5^(n + 4))
This simplifies to:
(n + 3) / ((n + 2) * 5)
Let's now compute the limit of this ratio as n approaches infinity:
lim n->[infinity] (n + 3) / ((n + 2) * 5)
When you complete the above operation, the result is 1/5.
For the ratio test, if the limit of the ratio is less than 1, this means the series converges. If the limit is more than 1, then the series diverges. If it equals to 1, the test is inconclusive.
Since our limit, 1/5, is less than 1, we can conclude that our series converges.