Determine whether the sequence converges or diverges. If it converges, find the limit.an = (1 + (6/n))^n
To determine whether the sequence an = (1 + (6/n))^n converges or diverges, we can use the concept of a limit.
As n approaches infinity (i.e., as n becomes larger and larger), let's evaluate the expression (1 + (6/n))^n. We can do this by rewriting the expression in the form of a limit:
lim(n->∞) (1 + (6/n))^n
To simplify the expression, we can rewrite it using the known limit of Euler's number (e) as n approaches infinity:
lim(n->∞) [(1 + (6/n))^n] = e^6
Thus, the limit of the sequence an = (1 + (6/n))^n as n approaches infinity is e^6.
Since the expression approaches a finite value, e^6, as n becomes increasingly larger, we can conclude that the sequence converges.
To determine whether the sequence converges or diverges, we need to analyze the behavior of the terms as n approaches infinity. In this case, the sequence is given by:
an = (1 + (6/n))^n
To find the limit of this sequence as n approaches infinity, we can rewrite it in exponential form using the concept of natural logarithms:
lim (n -> ∞) (1 + (6/n))^n = lim (n -> ∞) e^(n * ln(1 + (6/n)))
Now, let's break this down further:
lim (n -> ∞) e^(n * ln(1 + (6/n))) = e^(lim (n -> ∞) (n * ln(1 + (6/n))))
Now, we can focus on finding the limit of (n * ln(1 + (6/n))) as n approaches infinity:
lim (n -> ∞) (n * ln(1 + (6/n)))
We can simplify this expression by applying the property of limits:
lim (n -> ∞) (n * ln(1 + (6/n))) = lim (n -> ∞) (ln(1 + (6/n)) / (1/n))
We can now use L'Hôpital's Rule to evaluate this limit:
lim (n -> ∞) (ln(1 + (6/n)) / (1/n)) = lim (n -> ∞) ((1 / (1 + (6/n))) * ((-6/n^2) / -1))
Simplifying further:
lim (n -> ∞) ((1 / (1 + (6/n))) * ((-6/n^2) / -1)) = lim (n -> ∞) (6 / (n * (1 + (6/n))))
Now, as n approaches infinity, (6/n) approaches zero. Therefore, we can simplify the expression further:
lim (n -> ∞) (6 / (n * (1 + (6/n)))) = lim (n -> ∞) (6 / n)
Finally, taking the limit as n approaches infinity, we have:
lim (n -> ∞) (6 / n) = 0
Hence, the limit of the sequence an = (1 + (6/n))^n as n approaches infinity is 0. Therefore, the sequence converges to 0.
this might help:
https://planetmath.org/convergenceofthesequence11nn