Logarithm of Infinite Product of Real Numbers
Jump to navigation
Jump to search
Theorem
Let $(a_n)$ be a sequence of strictly positive real numbers.
Convergence
The following statements are equivalent:
- $(1): \quad$ The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$.
- $(2): \quad$ The series $\ds \sum_{n \mathop = 1}^\infty \ln a_n$ converges to $\ln a$.
Divergence to zero
The following are equivalent:
- The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$.
Divergence to infinity
The following statements are equivalent::
- $(1): \quad$ The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ diverges to $+\infty$.