Prove from first principles that the sequence (an) is convergent, wherean = (−1)n + n2n + 1for all n.
Question
Solution 1
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:
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