Riemann sum problem. Need to represent in defined integral

Question

When $f(x)={\frac{1}{x}}$, compute the limit

$$\lim_{n\to\infty}\left(\sum_{k=1}f(1+{\frac{2k}{n}})\right){\frac{2}{n}}$$

I solved as $$\int_1^{3} \frac{1}{x} \, dx$$

Is this right?

Answer 1

If the function $f $ is integrable at $[a,b] $ then $$\lim_{n\to+\infty}\frac {b-a}{n}\sum_{k=1}^nf (a+k\frac {b-a}{n})=\int_a^bf (x)dx $$

In your case, $$a=1 \;,\;b=3$$ thus the limit is

$$\int_1^3\frac {dx}{x}=\ln (3) $$