Determine whether the sequence converges, and if so find its limit.
{ln n/ n} +∞
n=1
I know it converges to 1, but how?
To determine if the sequence {ln n/n} converges, we can use the limit definition of convergence.
Let's compute the limit as n approaches infinity:
lim(n->∞) ln(n)/n
Since this is an indeterminate form (ln ∞/∞), we can use L'Hôpital's rule to find the limit.
Differentiating the numerator and the denominator with respect to n gives:
lim(n->∞) 1/n / 1 = lim(n->∞) 1/n
Now, the limit is of the form 1/∞, which is equal to zero. Therefore, we can conclude that:
lim(n->∞) ln(n)/n = 0
Thus, the sequence {ln n/n} converges to zero as n approaches infinity, not 1.
To determine whether the sequence converges and find its limit, we should analyze the given sequence {ln n / n} as n approaches infinity.
1. Start by observing the behavior of the sequence as n increases. Plug in some values for n to get an idea of how the terms change.
For example, when n=1, the term becomes ln(1)/1 = 0/1 = 0.
When n=2, the term becomes ln(2)/2 ≈ 0.34657.
When n=10, the term becomes ln(10)/10 ≈ 0.23025.
Notice how the terms are decreasing as n increases.
2. Next, consider the natural logarithm function ln(x). The natural logarithm is a strictly increasing function, which means ln(x1) < ln(x2) if x1 < x2.
In our case, we have ln(n)/n for the terms in the sequence. As n grows larger, the denominator n increases faster than the numerator ln(n).
Therefore, the terms of the sequence {ln n / n} will approach zero as n goes to infinity.
3. So, based on our observations, we can conclude that the given sequence converges to zero.
Now to prove this mathematically, we will use the limit definition:
lim(n→∞) (ln n / n)
We can use L'Hôpital's Rule to evaluate this limit. Differentiate the numerator and denominator with respect to n:
lim(n→∞) [d/dn (ln n) / d/dn (n)]
lim(n→∞) [1/n / 1] (Applying the derivative of ln n = 1/n)
The limit simplifies to:
lim(n→∞) (1 / n) = 0
Therefore, the limit of the sequence {ln n / n} as n approaches infinity is 0.
Hence, you mentioned that the limit is 1, but based on the analysis, we find that the limit is actually 0.