Question

Use the definition to determine whether {(-1)^n} is a Cauchy sequence ?

Answer 1

i know it is not a Cauchy sequence because it is not converges. But how we show it by definition of Cauchy theorem?

Answer 2

anyone?

Answer 3

The idea of a Cauchy sequence is that the difference between numbers gets arbitrarily small far enough into the sequence.
So formally, for some N we have for all n>N: \[|a_n-a_{n-1}|<\epsilon\]
for any epsilon
Does that happen here?

Answer 4

what thomas said. also it is important to know how to negate the definition. a sequence is NOT cauchy if...

Answer 5

if \[\left[a _{n}-a _{n-1} \right]>\epsilon\] for some n>N

Answer 6

cauchy if
for all \(\epsilon>o\) there is an N such that for all m, n> N \(|a_n-a_m|<\epsilon\)

Answer 7

to negate the definition, change "for all" to "there exists" and negate the conclusion

sequence is not cauchy if
there exists an \(\epsilon\) such that for all N if m, n > N we have
\(|a_n-a_m|>\epsilon\)

Answer 8

you get to pick the epsilon, and i think \(\frac{1}{2}\) would work nicely

Answer 9

yeah..that's correct...i will try it with epsilon 1/2..

Answer 10

to be more precise i should have written
not cauchy if there exists an \(\epsilon\) such that for all N there exists n,m > N with \(|a_n-a_n|>\epsilon\)