Find the condition of $x$ and $y$ in order that $\prod_{n=1}^{\infty}(x^{n}+y^{n})$ converge. ($x$, $y\in R$)
Condition For Infinite Product's Convergence
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real-analysis
1 Answers
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In general, the infinite product $\prod_{n=1}^\infty a_n$ converges (which, by convention, means converges to a nonzero value) if and only if the series $\sum_{n=1}^\infty \log(a_n)$ converges. I'll do one case: $0 < x < y$. Then $x^n + y^n = y^n (1 + (x/y)^n)$ so $\log(x^n + y^n) = n \log y + \log(1+(x/y)^n) = n \log y + O((x/y)^n)$. Clearly we need $\log y = 0$, i.e. $y = 1$, and then since $\sum_n (x/y)^n$ converges this is sufficient too.
The other cases are left for you...