Prove that sequence (-1)^{n} is not convergent .

Question
Sequences
asked 2021-03-02
Prove that sequence \((-1)^{n}\) is not convergent .

Answers (1)

2021-03-03
If this sequence converged, then every subsequence will converge to the same limit. However notice that this sequence goes as follows:
\(1,−1,1,−1,1,−1,...\) As \(n=0,1,2,3,4,5,...\)
Therefore, notice that for even nn values, we have the subsequence 1,1,1,1,1,...1
which obviously converges to 1, while for odd nn values, we have the subsequence
−1,−1,−1,−1,..., which obviously converges to −1.
herefore, since both subsequences converge to different values (1 and −1), we have that this sequence does not converge.
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