How to find a limit of: $$ lim_{n\to\infty}\frac{\sqrt[n]{\prod_{i=0}^{n-1} (a+ib)}}{\sum_{i=0}^{n-1} (a+ib)} $$ where $a>0$ and $b>0$
The task context is Infinite Product. Please help me or just give me a tip on how to simplify the expression and approach the task.
Let use arithmetic and geometric mean inequality: $$ lim_{n\to\infty}\frac{\sqrt[n]{\prod_{i=0}^{n-1} (a+ib)}}{\sum_{i=0}^{n-1} (a+ib)} < lim_{n\to\infty}\frac{\sum_{i=0}^{n-1} (a+ib)/n}{\sum_{i=0}^{n-1} (a+ib)} = lim_{n\to\infty} \frac{1}{n} = 0 $$ Or equivalency we can replace the denominator: $$ lim_{n\to\infty}\frac{\sqrt[n]{\prod_{i=0}^{n-1} (a+ib)}}{\sum_{i=0}^{n-1} (a+ib)} < lim_{n\to\infty}\frac{\sqrt[n]{\prod_{i=0}^{n-1} (a+ib)}}{n\sqrt[n]{\prod_{i=0}^{n-1} (a+ib)}} = lim_{n\to\infty} \frac{1}{n} = 0 $$ Notice that the equality hold iff $n=1$.