# Limit of a function that is O(1)

Suppose $\operatorname{Var}(X_i) = \sigma^2_i$ and let $S_n = \sum_{i=1}^n \sigma^2_i = O(1)$, then I need to find $\lim_{n \to \infty} S_n$. Do we have $S_n \to \infty$?

• Hello! You may find it helpful to look at the Mathjax formatting guide to learn how to write mathematical symbols here. – user296602 Feb 3 '16 at 20:31
• I made some edits, trying to minimize the wording edits. – Ian Feb 3 '16 at 20:33
• Also, this is not really tractable; all your hypothesis implies is that $\lim_{n \to \infty} S_n$ is a finite number given by $\sum_{i=1}^\infty \sigma_i^2$. But this limit could potentially be any nonnegative number. – Ian Feb 3 '16 at 20:34

If $\sigma(i)={1\over i}$, $S_n=O(1)$ but $\lim_{n\to\infty} S_n={\pi^2\over6}$. So the limit may not be $+\infty$.