I'm studying a book on integration (Lebesgue Integration on Euclidean Space by Frank Jones) and am having trouble with the following problem:
Define the open set $$ G = \{(x,y)\in \mathbb{R}^2: 1 < x \text{ and } 0 < y < x^{-a}\} $$ where $a$ is a real number satisfying $a > 1$. Prove that $\lambda(G) = \frac{1}{a-1}$, where $\lambda$ is defined as \begin{multline} \lambda(G) = \sup\{\lambda(P): P \subset G, \; \\P \text{ is a finite union of non-degenerate rectangles of the form $[a_1,b_1] \times [a_2,b_2]$ }\} \end{multline}
My first thought was to approximate $G$ using partitions. I constructed the rectangles $$ I_{kn} = \left[\frac{k}{n},\frac{k+1}{n}\right] \times \left[(n+1)^{-a},\left(\frac{k+1}{n-1}\right)^{-a}\right],k=2, \cdots,n^2-1 $$ and defined $P_n = \bigcup_{k=2}^{n^2-1}I_{kn}$. Then $$ \lambda(P_n) = \sum_{k=2}^{n^2-1} \frac{1}{n}\left(\left(\frac{k+1}{n-1}\right)^{-a}-(n+1)^{-a}\right) $$ However, I wasn't able to find a closed form solution to this sum (except in special cases, such as when $a$ is a positive integer). Is there a different approach I should be taking?