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

Answer 1

Yes.

Let #l = lim_(n->+oo)a_n#. #a_n# is monotone so #b_n# will be monotone as well, and #lim_(n->+oo)b_n = lim_(n->+oo)(a_n)^2 = (lim_(n->+oo)(a_n))^2 = l^2#.
It's like with functions : if #f# and #g# have a finite limit at #a#, then the product #f.g# will have a limit at #a#.
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Answer 2

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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Answer from HIX Tutor

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.

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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