To what limit does the sequence Sn = (3+n) /3^n converge as n approaches infinity?

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.