Q. 32 - Calculus

Question

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

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

Step-by-Step Solution

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 1: Given information

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

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

Rewrite the given summation in expanded form. 

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

Step 3: 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 $14.411$.