If we write the second product as a partial product, then you get:
P(n) = 1/2 * 2/3 * 3/4 * … * (n-1)/n
This cancels to get P(n) = 1/n, so taking the limit gives the infinite product as 0.

The third product is the inverse of the second, so the partial product is P(n) = n. This doesn’t converge, so the infinite product diverges.

For the second one, you can imagine sliding all the numbers in the denominator over one term to the right to obtain the top series, but with an extra term on the left and right of the series equal to 1/n where ‘n’ is however far you take the series. So this one is equal to zero because you can make 1/n arbitrarily small for a large enough n.

In the third one, you can imagine sliding the numerators to the right to obtain the first series but now with an extra (n/1) term left over. Making ‘n’ as big as you want means that this series taken to infinity is infinitely large.

Do I win a cake? :)
Bryan
South Africa

Hmmm… I don’t see where you get your smile from :D