How does $\lim\limits_{n\to\infty} \sum_{i=1}^{n^2} 1$ result into 1+3+5... How can $\lim\limits_{n\to\infty} \sum_{i=1}^{n^2} 1$ result into 1+3+5...? Why odd numbers?

Hoo boy. You're in for a treat; this theorem was known to Pythagoras and his disciples.

Theorem: The sum of the first $n$ odd numbers is equal to $n^2$.

Proof: By picture.

The original image is part of a page containing two proofs of this identity: http://www.9math.com/book/sum-first-n-odd-natural-numbers. It is not mine.
