Question

Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) an = ln(n + 3) − ln(n)

Answer 1

$\ln(n+3)-\ln n=\ln\dfrac{n+3}n=\ln\left(1+\dfrac3n\right)$

We can show the sequence is bounded and monotonic.

Boundedness: $1+\dfrac3n>1$ for all $n>0$, so $\ln\left(1+\dfrac3n\right)>\ln1=0$ for all $n>0$.

Monotonicity: Consider the function

$f(x)=\ln\left(1+\dfrac3x\right)$

which has derivative

$f'(x)=\dfrac{1+\frac3x}{-\frac3{x^2}}=-\dfrac3{x^2+3x}$

which has negative sign for all $x>0$, and so $f(x)$ is strictly decreasing.

$\ln(n+3)-\ln n$ is bounded and monotonic, so the sequence converges.

As $n\to\infty$ we have $\dfrac3n\to0$, leaving us with the limit

$\displaystyle\lim_{n\to\infty}(\ln(n+3)-\ln n)=\lim_{n\to\infty}\ln\left(1+\frac3n\right)=\ln\left(1+\lim_{n\to\infty}\frac3n\right)=\ln1=0$