I need to evaluate the limits of these sequences and am having a very difficult time. Some direction would be greatly appreciated:
(a) a_n = ((n-3)/2)^n
I found lim(1-3n^-1)^n = 1, so I think this is convergent, but is this the right solution? I worry 1^(infinity) is an indeterminate form, is this correct? If so, what alternative strategy should I use? The root test was inconclusive for me. Can I compare it to another function b_n? If so what function?
(b) a_n = (n!)^2 / (2n)!
I usually use the Ratio test to find out if the series is convergent for this kind of problem, but my ratio test results in a limit of 1, inconclusive. I'm not sure how to evaluate the limit of a problem like this directly.
(c) a_n = sqrt(n^2 + 3n) - n
I'm completely lost on this one.
Thanks very much!
(a) a_n = ((n-3)/2)^n
I found lim(1-3n^-1)^n = 1, so I think this is convergent, but is this the right solution? I worry 1^(infinity) is an indeterminate form, is this correct? If so, what alternative strategy should I use? The root test was inconclusive for me. Can I compare it to another function b_n? If so what function?
(b) a_n = (n!)^2 / (2n)!
I usually use the Ratio test to find out if the series is convergent for this kind of problem, but my ratio test results in a limit of 1, inconclusive. I'm not sure how to evaluate the limit of a problem like this directly.
(c) a_n = sqrt(n^2 + 3n) - n
I'm completely lost on this one.
Thanks very much!
