Question

Hi, I am wondering whether the sequence cos nπ/2 converges, diverges, or alternates between -1 and 1? thanks

Answer 1

It alternates.

Answer 2

how about this one? thanks
[-1+((-1)^{n ^{2}}/ n ^{2})\]

Answer 3

what happens as n gets really, really large?

Answer 4

an gets close negative 1

Answer 5

yup. there you go.

Answer 6

so diverges eh?

Answer 7

uh, no... it converges to -1...

Answer 8

so if n gets larger, and an approaches 0 or a number, the series converge?

Answer 9

yes

Answer 10

thanks a lot mate

Answer 11

sorry to disturb again, I thought that if lim n-infinity, and an approaches only 0, it is a convergence series, and if approaches infinity or any real number it is a dvergence series, Is there any exception regarding this theory?

Answer 12

Oh, I'm sorry. I misunderstood your question. The sequence converges if
\[ \lim_{n\rightarrow \infty} a_n = L\]
where L is any real number. The convergence of a SERIES is more complex. If a series is to converge, then
\[ \lim_{n\rightarrow \infty} a_n = 0 \]
but this is not the only requirement.

Answer 13

okay, last thing what is the requirement for a sequence ( not of a series) to diverge?? thank a lot

Answer 14

A sequence diverges if the limit as n -> infinity doesn't exist. Like,
\[ a_n = \log(n) \]