This is not a homework type question, I simply wish to understand a concept which is better demonstrated by this question:
Let$$S_n=\sum_{k=1}^n\frac n{n^2+kn+k^2},\quad T_n=\sum_{k=0}^{n-1}\frac n{n^2+kn+k^2}.$$
Now I know how to solve this equation by converting limit sum to integral and taking $\frac kn=x$ and I get $\frac{\pi}{3\sqrt3}$ as the answer, but the answer in the question is all inequalities for example less than or greater to $\frac{\pi}{3\sqrt3}$ and I learnt after searching the internet a little that this is an example of upper and lower sum. The answer for the given question is $S_n$ less than $\frac{\pi}{3\sqrt3}$ and $T_n$ greater than $\frac{\pi}{3\sqrt3}$.
Now I am wondering if anyone could explain to me why the rectangles formed while integrating the function in the graph of the areas of the functions are different and why is one above the graph and other below the graph. also for $T_n$ shouldnt the area be started from $x=\frac 1n$ some of the solutions I saw had the Graph for $T_n$ above the function $\frac{1}{x^2+x+1}$ and starting from $0$. Could anyone explain this function to me?