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Number Theory

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Compare Two Double Infinite Products

NUMTH-GJRJBS

Which answer is true when comparing $\prod_{m = 1}^{\infty}\prod_{n = 1}^{\infty}ne^{-m}$ and $\prod_{n = 1}^{\infty}\prod_{m = 1}^{\infty}ne^{-m}$?

A

$\prod_{m = 1}^{\infty}\prod_{n = 1}^{\infty}ne^{-m} = \prod_{n = 1}^{\infty}\prod_{m = 1}^{\infty}ne^{-m}$

B

$\prod_{m = 1}^{\infty}\prod_{n = 1}^{\infty}ne^{-m} < \prod_{n = 1}^{\infty}\prod_{m = 1}^{\infty}ne^{-m}$

C

$\prod_{m = 1}^{\infty}\prod_{n = 1}^{\infty}ne^{-m} > \prod_{n = 1}^{\infty}\prod_{m = 1}^{\infty}ne^{-m}$

D

$\prod_{m = 1}^{\infty}\prod_{n = 1}^{\infty}ne^{-m}$ and $\prod_{n = 1}^{\infty}\prod_{m = 1}^{\infty}ne^{-m}$ cannot be compared