- #1
sabyakgp
- 4
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Hello Friends,
I am at a loss to understand a proof concerning the proof of divergence of (-1) ^n sequence.
According to the book:
"To prove analytically that the sequence is convergent, it must satisfy both of the following conditions:
A: |-1-L| < epsilon
B: |+1 - L| < epsilon
"
(+1 and -1 are the only two values the sequence (-1)^n)
But, the book goes on, if we suppose epsilon = 1/2
"|-1-L| <1/2
which will not hold if L >0 since in that case |-1-L| = 1+L which is greater than 1/2
Likewise,
|+1-L| < 1/2 will not hold if L<0 since |+1-L| = 1 + |L| which is greater than 1/2.
Therefore since L can not be both negative and positive, there can be no limit to this sequence.
"
Though I understand the conditions, but fail to understand how it was proved that the limit does not exist. Could anyone please help me understand this?
Best Regards,
Sabya
I am at a loss to understand a proof concerning the proof of divergence of (-1) ^n sequence.
According to the book:
"To prove analytically that the sequence is convergent, it must satisfy both of the following conditions:
A: |-1-L| < epsilon
B: |+1 - L| < epsilon
"
(+1 and -1 are the only two values the sequence (-1)^n)
But, the book goes on, if we suppose epsilon = 1/2
"|-1-L| <1/2
which will not hold if L >0 since in that case |-1-L| = 1+L which is greater than 1/2
Likewise,
|+1-L| < 1/2 will not hold if L<0 since |+1-L| = 1 + |L| which is greater than 1/2.
Therefore since L can not be both negative and positive, there can be no limit to this sequence.
"
Though I understand the conditions, but fail to understand how it was proved that the limit does not exist. Could anyone please help me understand this?
Best Regards,
Sabya