(a) The series = ?
b) The sequence = ?
I have done the rest of the problems with out any problem.. but i cant seem to get this one right.
Please help,
Thank you
Need to think a bit about part a), but for now, here is some help with part b) if you set x = 2 / k (then note that x goes to zero as k goes to infinity.
The typical term of the sequence becomes A_k = (1 + x) ^(2/x) =
= [(1+x)^(1/x)]^2 whose limit, as x goes to zero, is e^2, where e is Euler's Constant. Some numerical investigation in Excel also verifies:
k A_k
1 3
2 4
3 4.62962963
4 5.0625
5 5.37824
6 5.618655693
7 5.8077951
10 6.191736422
15 6.536796047
100 7.244646118
1000 7.37431239
10000 7.387578632
Good luck!!