If #a_n# converges and #a_n >b_n# for all n, does #b_n# converge?

Answer 1

See explanation below

We assume that we are talking about sequences (although, for infinite series, the reasoning is the quite similar)

Assume that #{a_n}# converges. This is the same #lim_(ntooo)a_n=L# with #L# a finite number
For all #{b_n}# we know that #b_n < a_n#
If #{b_n}# is a growing sequence, then all terms are under the value #L# and #{b_n}# converges perhaps to #L#
If #{b_n}# is a decreasing or alternate sequence, we can´t assume the convergence. Think in this example
#a_n=3+1/n# and #b_n=(-1)^n#
In this case #b_n < a_n# for all #n#, but #lim_(ntooo)a_n=3# and #lim_(ntooo)b_n# doesn`t exist
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Answer 2

Yes, if (a_n) converges and (a_n > b_n) for all (n), then (b_n) also converges.

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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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