Question

Write out each sum in Exercises 32 in expanded form, and then calculate the value of the sum.

$\sum _{k=3}^{10}\ln k$

Short answer

The expanded notation is$\begin{array}{rcl}& & \ln \left(3\right)+\ln \left(4\right)+\ln \left(5\right)+\ln \left(6\right)+\ln \left(7\right)+\ln \left(8\right)+\ln \left(9\right)+\ln \left(10\right)\end{array}$ and their sum is $\ln \left(1814400\right)$or $14.411$.

Step-by-step solution

Given information

The given summation is $\sum _{k=3}^{10}\ln k$.

Write the summation in expanded form and evaluate the sum. 

Rewrite the given summation in expanded form.

$\begin{array}{rcl}\sum _{k=3}^{10}\ln k& =& \ln \left(3\right)+\ln \left(4\right)+\ln \left(5\right)+\ln \left(6\right)+\ln \left(7\right)+\ln \left(8\right)+\ln \left(9\right)+\ln \left(10\right)\\ & =& \ln \left(3·4·5·6·7·8·9·10\right)[\ln \left(m\right)+\ln \left(n\right)=\ln \left(mn\right)]\\ & =& \ln \left(1814400\right)\\ & =& 14.411\end{array}$

Write the conclusion

The expanded form is $\begin{array}{rcl}& & \ln \left(3\right)+\ln \left(4\right)+\ln \left(5\right)+\ln \left(6\right)+\ln \left(7\right)+\ln \left(8\right)+\ln \left(9\right)+\ln \left(10\right)\end{array}$.

The sum is$\ln \left(1814400\right)$or$14.411$.