3. Determine whether the sequence $$ a_{n}=\frac{x+3}{\ln x} $$ converges or diverges. If it converges, find its limit.

Question

UpStudy AI · Accepted Answer

1. We interpret the sequence as $$ a_n=\frac{n+3}{\ln n} $$ where $$ n $$ is a positive integer.
2. As $$ n \to \infty $$, the numerator $$ n+3 $$ grows linearly, while the denominator $$ \ln n $$ grows much slower (logarithmically).
3. To analyze the limit, consider

$$
   \lim_{n \to \infty} \frac{n+3}{\ln n}.
   $$

For large $$ n $$, the behavior of $$ n+3 $$ is essentially the same as $$ n $$, so we examine

$$
   \lim_{n \to \infty} \frac{n}{\ln n}.
   $$

4. The numerator increases without bound, and since the logarithm increases very slowly, the ratio tends to infinity.
5. Thus, we conclude that

$$
   \lim_{n \to \infty} \frac{n+3}{\ln n} = \infty,
   $$

meaning that the sequence diverges (specifically, it diverges to infinity).
The sequence $$ a_n = \frac{n+3}{\ln n} $$ diverges to infinity as $$ n $$ approaches infinity.

Determine whether the sequence converges or diverges. If it converges, find its
limit.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Calculus Questions

Determine whether the sequence converges or diverges. If it converges, find its limit.