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To find the limit of the sequence Sn = (3+n)/3^n as n approaches infinity, we can use the concept of limits and apply some algebraic manipulation.
Let's break down the steps:
Step 1: Rewrite the expression of Sn to simplify it:
Sn = (3+n)/3^n = 3/3^n + n/3^n = 3/3^n + 1/3^(n-1)
Step 2: Notice that we have two terms in the expression for Sn:
a) The first term, 3/3^n, approaches 0 as n approaches infinity because any positive number divided by a larger power of 3 will tend to zero.
b) The second term, 1/3^(n-1), also approaches 0 as n approaches infinity because the denominator grows faster than the numerator.
Step 3: Since both terms approach 0 as n approaches infinity, their sum (Sn) will also approach 0.
Therefore, the limit of the sequence Sn = (3+n)/3^n as n approaches infinity is 0.