Summation of harmonic series.

Question

I'm trying to figure out how to answer this linear algebra question and can't figure it out. Can someone please explain it to me?

Thanks a bunch!

Here's the questions:

Answer 1

To do this consider the integral $$\int_{1}^{x}\frac{1}{t}dt.$$ We can approximate it from below and from above using lower and upper Riemann sums for the partition that cuts at the integers $1,2,3,...$.

Since $\frac{1}{t}$ is decreasing the lower and upper Riemann sums are obtained when the function is evaluated at the right boundary and the left boundary of the intervals of the partition.

We get $$\sum_{n=1}^{m} \frac{1}{n}=\sum_{n=1}^{m} \frac{1}{n}(n+1-n)>\int_{1}^{x}\frac{1}{t}dt=\ln(x)>\sum_{n=1}^{m} \frac{1}{n+1}(n+1-n)=\sum_{n=1}^{m}\frac{1}{n+1}$$