Prove from first principles that the sequence (an) is convergent, wherean = (−1)n + n2n + 1for all n.
Question
Prove from first principles that the sequence is convergent, where for all .
Solution
To prove that a sequence (an) is convergent, we need to show that it has a limit as n approaches infinity.
The sequence given is an = (−1)^n + n^2/(n + 1).
Let's first simplify the sequence:
an = (−1)^n + n^2/(n + 1) = (−1)^n + 1 - 1/n
Now, let's find the limit as n approaches infinity:
lim (n→∞) an = lim (n→∞) [(−1)^n + 1 - 1/n]
We can separate this into two limits:
= [lim (n→∞) (−1)^n] + [lim (n→∞) (1 - 1/n)]
The limit of (−1)^n as n approaches infinity does not exist because it oscillates between -1 and 1.
The limit of (1 - 1/n) as n approaches infinity is 1 because as n gets larger, 1/n gets closer to 0.
Therefore, the sequence (an) does not have a limit as n approaches infinity and is not convergent.
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