Determine whether the sequence converges or diverges. if it converges, find the limit. An=3^(n+2)/(5^n)

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The nth tern of the sequence is An = 3^(n+2)/5^n.

To determine the convergence of An.

Solution:

An = 3^(n+2)/5^n = 9(3/5)^n = 9x^n where x<1.

Taking limits 9x^n = 0 as approaches infinity, as x-->infinity limit x^n = 0 for all 0<=x <1.

So {An} is convergent.

An = [3^(n+2)]/(5^n)

= [3^(n+2)]/[(3*5/3)^n]

= [3^(n+2)]/[(3^n)*(5/3)^n]

= (3^2)/[(5/3)^n]

= (5/3)^n

As 5/3 is greater than 1, as value of n increases,

the value of (5/3)^n also increases,

and value of 9/(5/3)^n decreases.

Therefore, sequences An converges.

To find value of limit of An:

When n approaches infinity:

Value of limit (5/3)^n becomes infinity

and value of limit 9/(5/3)^n becomes:

9/(infinity) = 0

Answer:

Limit An = [3^(n+2)]/(5^n) as n approaches infinity is 0

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