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divergent sequence

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(x_n tends to plus infinity if and only if for each M belonging to real numbers there exists an N belonging to natural numbers such that:
n>=N implies x_>M.
x_n is said diverge yo minus infinity if and only if for each M belonging to natural numbers such that:
n>=N implies x_n <M.)

Prove by above definition that each of the following sequences diverge to plus infinity or minus infinity.

a) x_n = n -3n^2

b)x_n=(n^2+1)/n

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Solution Summary

This solution shows how to provide proof that the given sequences diverge to plus infinity or minus infinity based on a given set of definitions.

Solution Preview

(a) x_n diverges to -oo (minus infinity).
Proof: For each real number M, without the loss of generality, we only consider a very small M<-1
We ...

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