# Suppose, #a_n# is monotone and converges and #b_n=(a_n)^2#. Does #b_n# necessarily converge?

Yes.

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Yes, if ( a_n ) is a monotone sequence that converges, then ( b_n = (a_n)^2 ) will also converge. This is because if ( a_n ) is convergent, it means that it approaches a finite limit as ( n ) approaches infinity. When you square a convergent sequence, ( b_n = (a_n)^2 ), the resulting sequence will also converge to the square of the limit of ( a_n ). Thus, ( b_n ) will converge.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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