Limits and sequences math problem

[*FaerieLight*]

Homework Statement

Find the limit as n\rightarrow\infty of the sequence

a_n=\frac{(2n)! 2²ⁿ}{(n!)² (2n+1) 5²ⁿ}

Homework Equations

The Attempt at a Solution

 

[fzero]

Fixed the formatting, I think:

<br /> a_n = \frac{(2n)! 2^{2n} }{(n!)^2 (2n+1) 5^{2n}}.<br />

If that's right, what have you tried?

 

[*FaerieLight*]

I've just worked out how to do the problem. I found the ratio of a_n+1 to a_n, as n approaches infinity, and the ratio is less than 1 (16/25), so by the ratio test, this means the sequence approaches 0 as n approaches infinity. Thanks for fixing the formatting. Indeed the expression is correct.

 

[Mark44]

The ratio test is to be used on terms of an infinite series, \sum a_n. If the limit of the ratio a_(n + 1)/a_n is less than 1, the series converges. You should be able to evaluate the limit of this sequence directly.

The tricky part is evaluating (2n)!/[(n!)^2 (2n + 1)], since 2^(2n)/5^(2n) = (2/5)^(2n) --> 0 as n --> infinity.

 

[Dick]

The ratio test is to be used on terms of an infinite series, \sum a_n. If the limit of the ratio a_(n + 1)/a_n is less than 1, the series converges. You should be able to evaluate the limit of this sequence directly.

The tricky part is evaluating (2n)!/[(n!)^2 (2n + 1)], since 2^(2n)/5^(2n) = (2/5)^(2n) --> 0 as n --> infinity.

You can also use the ratio test on a sequence. If |a_(n+1)/a_n| goes to a limit less than 1 then the sequence converges to zero. As it must if the series is to converge. For some reason this doesn't seem to be taught much as a method.

 

[Mark44]

You can also use the ratio test on a sequence. If |a_(n+1)/a_n| goes to a limit less than 1 then the sequence converges to zero. As it must if the series is to converge. For some reason this doesn't seem to be taught much as a method.

I don't remember learning about using the ratio test on a sequence, or seeing it in any of the calculus texts I taught out of.

 

[Dick]

I don't remember learning about using the ratio test on a sequence, or seeing it in any of the calculus texts I taught out of.

No, I don't think I have either. But it works, right?