Q. 31 - Calculus

Question

Write out each sum in Exercises 31 in expanded form, and then calculate the value of the sum.

Step-by-Step Solution

Answer

The expanded notation is $0+\frac{1}{8}+\frac{1}{2}+\frac{9}{8}+2+\frac{25}{8}$ and their sum is $\frac{55}{8}$ or $6.875$.

Step 1: Given information

The given summation is $\sum _{k=0}^5{\left(\frac{1}{2}k\right)}^2\left(\frac{1}{2}\right)$.

Step 2: Write the summation in expanded form and evaluate the sum.

Rewrite the given summation in expanded form.

$\begin{array}{rcl}\sum _{k=0}^5{\left(\frac{1}{2}k\right)}^2\left(\frac{1}{2}\right)& =& {\left(\frac{1}{2}\left(0\right)\right)}^2\left(\frac{1}{2}\right)+{\left(\frac{1}{2}\left(1\right)\right)}^2\left(\frac{1}{2}\right)+{\left(\frac{1}{2}\left(2\right)\right)}^2\left(\frac{1}{2}\right)+{\left(\frac{1}{2}\left(3\right)\right)}^2\left(\frac{1}{2}\right)+{\left(\frac{1}{2}\left(4\right)\right)}^2\left(\frac{1}{2}\right)+{\left(\frac{1}{2}\left(5\right)\right)}^2\left(\frac{1}{2}\right)\\ & =& 0+\frac{1}{4}\left(\frac{1}{2}\right)+\frac{1}{2}+\left(\frac{9}{4}\right)\left(\frac{1}{2}\right)+\left(4\right)\left(\frac{1}{2}\right)+\left(\frac{25}{4}\right)\left(\frac{1}{2}\right)\\ & =& 0+\frac{1}{8}+\frac{1}{2}+\frac{9}{8}+2+\frac{25}{8}\\ & =& \frac{35}{8}+\frac{5}{2}\\ & =& \frac{35+20}{8}\\ & =& \frac{55}{8}\\ & =& 6.875\end{array}$

Step 3: Write the conclusion

The expanded form is $0+\frac{1}{8}+\frac{1}{2}+\frac{9}{8}+2+\frac{25}{8}$.

The sum is $\frac{55}{8}$ or $6.875$.