Converging sequences

[mahxirb]

Any help showing that the following sequences converge and what their limits are would be appreciated:

1) t_n, where t_0 = 1 and t_n+1 = sqrt(a+t_n), with a > 0 and is fixed.

2) (n^k)/((1+1/k)^n), with k a natural number and fixed.

 

[stapel]

Any help showing that the following sequences converge and what their limits are would be appreciated:

1) t_n, where t_0 = 1 and t_n+1 = sqrt(a+t_n), with a > 0 and is fixed.

Did you try picking a number for "a" and seeing what happened? Did you then try working with the formula?

2) (n^k)/((1+1/k)^n), with k a natural number and fixed.

Eventually (since k is fixed), n will be larger than k. Start with that assumption: n > k. What can you do with this?

 

[HallsofIvy]

Any help showing that the following sequences converge and what their limits are would be appreciated:

1) t_n, where t_0 = 1 and t_n+1 = sqrt(a+t_n), with a > 0 and is fixed.

If this converges, to "T", say, then, taking the limit of both sides of \(\displaystyle t_{n+1}= \sqrt{a+ t_n}\) we must have
\(\displaystyle T= \sqrt{a+ T}\). Solve that for T. Then use that value of T to show this does converge.

2) (n^k)/((1+1/k)^n), with k a natural number and fixed.