 Problem Statement

1. Show that ##\sum_{n=1}^N \arctan{(n)} \geq N \arctan{(N)}(1/2)\ln{(1+N^2)}##
2. Determine numbers ##\alpha##, ##\beta## such that ##\lim\limits_{N\to\infty} N^\alpha \sum_{n=1}^N \arctan{(n)} =\beta## with ##\beta \neq 0##
 Relevant Equations
 ##\int_a^b f(x) dx= F(b)F(a)##
1. ##\sum_{n=1}^N \arctan{(n)} \geq N \arctan{(N)}(1/2)\ln{(1+N^2)} \iff \sum_{n=1}^N \arctan{(n)} \geq N \int_0^N \frac{1}{1+x^2} dx  \int_0^N \frac{x}{1+x^2} dx##
Where do I go from here? I've tried understanding this graphically, but to no avail.
2. Maybe this follows from finding an upper and lower bound for ##N^\alpha \sum_{n=1}^N \arctan{(n)}##, and then somehow applying the squeeze theorem?
Where do I go from here? I've tried understanding this graphically, but to no avail.
2. Maybe this follows from finding an upper and lower bound for ##N^\alpha \sum_{n=1}^N \arctan{(n)}##, and then somehow applying the squeeze theorem?