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If there are $\prod_{i=1}^{\infty} i$ elements in a set where $i$ is a natural number, does the cardinality of a set equal to $\aleph_0$?

Also, if there are $\Sigma_{i=1}^{\infty} 2^{i} $ elements in a set, what does the cardinality of a set equal to?

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    The product has cardinality continuum.2012-02-15
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    @azarel Thank you. I edited a question to add one more question. Can you also help me here? Thanks.2012-02-15
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    $\sum_{i<\aleph_0}2^i=\aleph_0$ if that is what you mean.2012-02-15

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