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How do you determine whether the sequence #a_n=(2^n+3^n)/(2^n-3^n)# converges, if so how do you find the limit?

Answer 1

Paisley Johnson

#lim_(n->oo) a_n = -1#

Divide both numerator and denominator by #3^n# like this...

#lim_(n->oo) a_n = lim_(n->oo) (2^n+3^n)/(2^n-3^n)#

#color(white)(lim_(n->oo) a_n) = lim_(n->oo) ((2/3)^n+1)/((2/3)^n-1)#

#color(white)(lim_(n->oo) a_n) = (0+1)/(0-1)#

#color(white)(lim_(n->oo) a_n) = -1#

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Answer 2

Scarlett Allison

To determine whether the sequence (a_n = \frac{2^n + 3^n}{2^n - 3^n}) converges, we analyze the behavior as (n) approaches infinity.

When (n) approaches infinity, the term (3^n) dominates the sequence because it grows much faster than (2^n). Consequently, the term (3^n) dominates both the numerator and the denominator. Thus, the sequence behaves as (\frac{3^n}{3^n} = 1) as (n) approaches infinity.

Therefore, the sequence converges, and its limit as (n) approaches infinity is (1).

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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.