Help Needed: Understanding An=1x3x5... (2n-1)/(2n)^n Convergence

[anderma8]

I'm trying to figure out the following:

An=1x3x5... (2n-1)/(2n)^n and I'm trying to determine if it converges or diverges and if it converges, what the limit is. The answer is 1/2n x 3/2n x 5/2n... (2n-1)/2n and it converges, but I don't understand what they did or how they got to the answer. Any help?

 

[HallsofIvy]

I'm trying to figure out the following:

An={1x3x5... (2n-1)}/(2n)^n and I'm trying to determine if it converges or diverges and if it converges, what the limit is. The answer is 1/2n x 3/2n x 5/2n... (2n-1)/2n and it converges, but I don't understand what they did or how they got to the answer. Any help?

Your "answer" does not give the limit. What they have done, very simply, is take that (2n)^n in the denominator and distribute it to each of the factors in the numerator. For example, if n= 3 you have {1*3*5}/(6^3)= (1/6)(3/6)(5/6). Since each factor is less than 1, and multiplying any positive number by a number less than one gives a smaller product, multiplying by more and more such fractions makes the product smaller and smaller- the sequence clearly converges to 0.