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

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

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.

### Relevant Questions

Prove that sequence $$\displaystyle{\left(-{1}\right)}^{{{n}}}$$ is not convergent .
Several terms of a sequence {a_n}_(n=1)^oo are given. a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. {1, 3, 9, 27, 81, ...}
Prove by induction that $$\displaystyle{1}+{3}+⋯+{\left({2}{n}+{1}\right)}={\left({n}+{1}\right)}^{{2}}.$$
(a) Find a recurrence relation that defines the sequence 1, 1, 1, 1, 2, 3, 5, 9, 15, 26, ... (Hint: each number in the sequence is based on the four numbers just before in the sequence).
(b) Now find a different sequence that satisfies the recurrence relation you found in (a)