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 (an) is convergent, wherean = (−1)n + n2n + 1for all n.
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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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