Show that this sequence converges to 0

[fishingspree2]

Hello

I need to show that this sequence converges to 0:
http://img238.imageshack.us/img238/139/77726816.gif

Here is my work:
http://img238.imageshack.us/img238/2570/eqlatexlimnrightarrowin.gif
There is n factors in the numerator and in the denominator. We can pair each factor one with another:
http://img142.imageshack.us/img142/2570/eqlatexlimnrightarrowin.gif

As n goes to infinity the first terms of this limit go to 0, and thus the limit is = 0.

Is my proof acceptable? Can someone show me another way to do it?
Thank you

 

[mathman]

Your proof is faulty since there is no requirement that r be an integer. However you seem to have the general idea in that once n>r, the additional multipliers as n gets bigger are <1 and approach 0.

 

[fishingspree2]

Your proof is faulty since there is no requirement that r be an integer. However you seem to have the general idea in that once n>r, the additional multipliers as n gets bigger are <1 and approach 0.

thank you for your answer
can you suggest me another way to do it?

 

[mathman]

thank you for your answer
can you suggest me another way to do it?

Not offhand. The approach described seems to be the simplest.

 

[Werg22]

Notice that a*(a+1)*(a+2)*...*(a + n - 1) > a^n. So 1 / a*(a+1)*(a+2)*...*(a + n - 1) < 1/a^n. See where you can make use of that.