Does this continued fraction [1;1, 1/2, 1/3, 1/4, 1/5,...] converge to Pi/2?

Questions
1
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2

Yes. The formula is equivalent to a continued fraction expansion of Wallis’s formula that I believe is due to Stern (Google is not helping me confirm that at the moment). The convergents may be found here:

See the last remark under “Formula” at the OEIS links to see your CF formula.

3

I was disappointed by the output of this:

ContinuedFractionK[1, 1/n, {n, 1, \[Infinity]}] // N