Problem 27 - Essential Calculus

Question

In Exercises \(27-30,\) find the limit of \(s(n)\) as \(n \rightarrow \infty\) $$ s(n)=\frac{81}{n^{4}}\left[\frac{n^{2}(n+1)^{2}}{4}\right] $$

Step-by-Step Solution

Answer

So, the limit of the function \(s(n)\) as \(n \rightarrow \infty\) is \(20.25\).

Step 1: Simplify the Function

Rewrite \(s(n)\) as \[ s(n) = \frac{81n^{2}(n + 1)^{2}}{4n^{4}} \]Then simplify the function to: \[ s(n)= \frac{81(n + 1)^{2}}{4n^{2}} \]

Step 2: Divide each term by \(n^{2}\)

To find the dominant term to calculate the limit as \(n \rightarrow \infty\), divide each term in the numerator and denominator by \(n^{2}\)\[ s(n) = \frac{81(1 + 1/n)^2}{4}\]

Step 3: Taking the limit

Now, take the limit as \(n \rightarrow \infty\): \[ \lim_{n \rightarrow \infty} s(n) = \frac{81(1 + 0)^{2}}{4} \]