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Results tagged with sequences-and-series
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user 89985
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
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Prove that if $\sum_{t=1} ^{\infty} x_t = \infty$ where $x_t \in [0,1], \forall t$, then $\p... [closed]
Prove that if $\sum_{t=1} ^{\infty} x_t = \infty$ where $x_t \in [0,1], \forall t$, then $\prod_{t = 1}^{\infty} (1-x_{t}) = 0$.
Many thanks.
P/S:
Not homework but it is a part of the proof in a …
- 139
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Prove that if $\sum_{t=1} ^{\infty} x_t = \infty$ where $x_t \in [0,1], \forall t$, then $\p...
Sketch proof (relating the infinite product to the infinite sum via the inequality $e^x \geq 1 + x, \forall x$):
Since $x_t \leq -\log(1 - x_t), \forall t$ and $\sum_{t=1}^{\infty} x_t = \infty$, we …
- 139