Prove that \frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots-\frac{1}{2009}+\frac{1}{2010}<\frac{3}{8}
Oh my, I feel embarrased for not knowing how to solve such an elementary problem but I'm really stuck on this one. I mean - I tried grouping them (I mean - these pairs with minus in-between) to show that the first pair gives $\frac{1}{6}$ and from there it's descending so it has to be less than $\frac{3}{8}$ but it doesn't seem to be a good idea after all. So I tried to remodel the pair-thinking and noted that: $\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}$ which leads to pairing the initial sequence to the form of: $\frac{1}{2\cdot3}+\frac{1}{4\cdot5}+\frac{1}{6\cdot7}+\cdots+\frac{1}{2008\cdot2009}+\frac{1}{2010}$ but it still seems to be leading nowhere. How should I approach it?