## Morris: Infinite Products

This made me smile.  I hope it makes you smile too.

What is

If you think they are the same then why?  If you think they are different then why?

1. ### tjmathematica said,

June 8, 2009 at 3:54 pm

a) lim(n->infinity) PRODUCT(n/n) = 1
b) lim(n->infinity) PRODUCT(n/(n+1)) < 1
value starts at 1/2 and decreses
c) lim(n->infinity) PRODUCT((n+1)/n) > 1
value starts at 2 and increses

2. ### Brian Tristam Williams said,

June 8, 2009 at 4:38 pm

Yeah, so a) trends toward 1 :) and b) trends toward 0 and c) trends toward inifinity [and beyond!] being that c) is the reciprocal of b) but…

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

3. ### Bryan said,

June 8, 2009 at 4:43 pm

The first series is obviously equal to 1 because each term is equal to 1.

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

4. ### Brian Tristam Williams said,

June 8, 2009 at 5:12 pm

Ok, never mind, lol I get it. “Infinite Products” picture pun. tyvm

5. ### Andy said,

June 8, 2009 at 5:22 pm

The first product is very clearly 1.

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.

6. ### Stephen Morris said,

June 8, 2009 at 6:44 pm

Brian, thanks for getting the picture pun, it was tricky getting David Beckham and James Bond to pose for me.  I’m hoping for Victoria and Moneypenny next time.

Actually what made me smile was the maths.  I love the fact that all of these products cancel completely [spoiler]and yet they all have completely different answers.[/spoiler]Unfortunately I can’t afford prizes but anyone who gets this right should reward themselves with a Gucci bag or a Ferrari.  Bryan, you should buy yourself a cake shaped like a Gucci bag or a Ferrari!
7. ### Mark said,

June 12, 2009 at 12:13 pm

a) 1
b) infinity
c) 1/infinity