# Find $\lim_{n \to \infty} \frac{x_n}{x_1 + \cdots + x_n}$

Suppose that $\{x_n\}_{n=1}^{\infty}$ is a bounded sequence, and that $x_n>0$ holds for all positive integer $n$.

Find $\lim_{n \to \infty} \frac{x_n}{x_1 + \cdots + x_n}$.

• Any ideas, guesses, thoughts, work? – blue Sep 2 '14 at 3:23
• Is it series or sequence you meant? – Anjan3 Sep 2 '14 at 3:52
• @AnjanDebnath does it affect your understanding? – John Fernley Sep 2 '14 at 4:17
• @JohnFernley Does my understanding affect your understanding? For the first time, when I saw the question it was " Suppose that $\{x_n\}_{n=1}^\infty$ is a bounded 'series' and now after the editing is done(after you made the comment), I see "bounded sequence". Do I need to elaborate more for understanding? – Anjan3 Sep 2 '14 at 5:10
• @AnjanDebnath If you understand the question don't distract from it with this pedantry. We weren't talking about my understanding. – John Fernley Sep 2 '14 at 21:07

Hint: It is useful to separately consider the cases $\sum x_n < \infty$ and $\sum x_n = \infty$. In the latter case, remember that $\{x_n\}$ is bounded.
• @Rene Do you mean the sum? The partial sums are monotonic increasing, so either they are unbounded and diverge to $+\infty$ or are bounded and hence have a limit. – blue Sep 2 '14 at 3:38
I think the limit is zero. The above hint is useful. For $\sum x_n<\infty$, $x_n\rightarrow 0$ ($n\rightarrow \infty$). In the case $\sum x_n=\infty$, the answer is obvious.