Determine if a series converges or diverges?
Determine if the series from 100 to infinity of (2-e^(1/n))^n^2 converges conditionally or absolutely, or diverges.
I tried the root test but that didn't work. Explanation would be helpful thank you!
Determine if the series from 100 to infinity of (2-e^(1/n))^n^2 converges conditionally or absolutely, or diverges.
I tried the root test but that didn't work. Explanation would be helpful thank you!
Comments
Using the Root Test:
r = lim(n→∞) |(2 - e^(1/n))^(n^2)|^(1/n)
..= lim(n→∞) (2 - e^(1/n))^n
Using logarithms,
ln r = lim(n→∞) n ln(2 - e^(1/n))
......= lim(n→∞) ln(2 - e^(1/n))/(1/n)
......= lim(t→0+) ln(2 - e^t)/t, letting t = 1/n
......= lim(t→0+) [-e^t/(2 - e^t)] / 1, by L'Hopital's Rule (0/0)
......= -1/2.
So, r = e^(-1/2).
Since r = e^(-1/2) < 1, we conclude that this series converges.
I hope this helps!