1. ## convergence help

For each positive integer n, let

{y_n}= 1 + 1/2 +...+1/n- the integral of (1/x) from 1 to n.

Prove that the sequence {y_n} converges.

2. Originally Posted by friday616
For each positive integer n, let

{y_n}= 1 + 1/2 +...+1/n- the integral of (1/x) from 1 to n.

Prove that the sequence {y_n} converges.
Proof of monotone decreasing: $y_{n-1}-y_n=\sum_{k=1}^{n-1}\frac{1}{k}-\sum_{k=1}^n\frac{1}{k}+\int_1^n\frac{1}{x}\,dx-\int_1^{n-1}\frac{1}{x}\,dx$ $=\int_{n-1}^n\frac{1}{x}\,dx-\frac{1}{n}\geq\int_{n-1}^n\frac{1}{n}\,dx-\frac{1}{n}=0$

To prove that $\{y_n\}$ is bounded below by $0$, if $P$ is the partition $\{x_1=1,x_2=2,...,x_n=n\}$, and $f(x)=\frac{1}{x}$, the upper sum $U(P,f)$ of the integral of $f(x)$ is $\sum_{k=1}^{n-1}\frac{1}{x_k}(x_{k+1}-x_k)=\sum_{k=1}^{n-1}\frac{1}{k}$. Since $\sum_{k=1}^n\frac{1}{k}>U(P,f)>\int_1^n\frac{1}{x} \,dx$, it follows that $y_n>0$ for all $n$.

Since $\{y_n\}$ is a monotonically decreasing sequence bounded below, it converges.

This is actually a famous limit and converges to the Euler-Mascheroni Constant, $\gamma$.

$\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right)=\gamma\approx0.577$

You're welcome.