Q. 40 - Calculus

Question

Find a formula for each of the sums in Exercises 40, and then use these formulas to calculate each sum for  $n=100,500$ and $1000$

Step-by-Step Solution

Answer

The formula of the given summation is $\frac{n^2+3n+5}{3n^2}$.

The sum when $n=100$ is $0.3435$.

The sum when $n=500$ is $0.33534$.

The sum when $n=1000$ is $0.334335$.

Step 1: Given information

The given summation is .$\sum _{k=1}^n\frac{k^2+k+1}{n^3}$

Step 2: Determine the formula for the given summation.

The sum can be written as: 

$$\begin{gathered} \sum _{k=1}^n\frac{k^2+k+1}{n^3}=\frac{1}{n^3}\sum _{k=1}^n\left(k^2+k+1\right) \\ =\frac{1}{n^3}\left(\sum _{k=1}^n{k}^2+\sum _{k=1}^{n}k+\sum _{k=1}^{n}1\right) \\ =\frac{1}{n^3}\left[\left(\frac{n(n+1)(2n+1)}{6}\right)+\frac{n(n+1)}{2}+n\right] \\ =\frac{n^2+3n+5}{3n^2} \end{gathered}$$

Step 3: Evaluate the sum for n = 100 , 500 and 1000

Substitute $100$ for $n$ in $\frac{n^2+3n+5}{3n^2}$.

$\frac{{\left(100\right)}^2+3\left(100\right)+5}{3{\left(100\right)}^2}=0.3435$

Substitute $500$ for $n$ in $\frac{n^2+3n+5}{3n^2}$.

$\frac{{\left(500\right)}^2+3\left(500\right)+5}{3{\left(500\right)}^2}=0.33534$

Substitute $1000$ for $n$ in $\frac{n^2+3n+5}{3n^2}$.

$\frac{{\left(1000\right)}^2+3\left(1000\right)+5}{3{\left(1000\right)}^2}=0.334335$

Step 4: Write the conclusion

The formula is $\frac{n^2+3n+5}{3n^2}$.

The sum when $n=100,500$ and $1000$ are $0.3435,0.33534$ and $0.334335$.