<s>[INST] Question Sequences and Series Prove that the sequence $(a_n)_{n \geq 1}$ defined by $$ a_n= \sum_{k=1}^{n} \frac{1}{k} - \ln(n+1)$$
 is increasing and bounded. It is on the study guide for my final exam, which is tomorrow so I am trying to figure it out. Thanks
Can someone please help?
 [/INST] The part that $a_n$ is increasing is already done. We focus on proving $a_n$ is bounded above. We have:
$a_n = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n} - \ln(n+1) = 1 + \dfrac{1}{2} + \cdots + \dfrac{1}{n} - \ln n - \ln\left(1+\frac{1}{n}\right) = 1 + \dfrac{1}{2} + \cdots + \dfrac{1}{n-1} - \ln n+ \left(\dfrac{1}{n} - \ln\left(1+\frac{1}{n}\right)\right)$. Apply the MVT for the function $f(x) = x - \ln(1+x)$ on $(0,x)$ we have: $x - \ln(1+x) = f'(c)\left(x-0\right) = \dfrac{cx}{1+c} < cx, 0 < c < x$. Let $x = \dfrac{1}{n} \Rightarrow \dfrac{1}{n} - \ln(1+\frac{1}{n}) < c\dfrac{1}{n} < \dfrac{1}{n^2}$. By repeating this pattern back to $\ln 2$ we have that $a_n < 1 + \dfrac{1}{2^2} + \cdots + \dfrac{1}{n^2} < \dfrac{\pi^2}{6}$, proving boundedness of $a_n$.
</s>