Chapter 12.1, Problem 52E

To determine

Calculate the sum of the equation.

Answer to Problem 52E

The summation of the equation is $0.153$ .

Explanation of Solution

Given: It is given in the question that the equation is ${\displaystyle \sum _{j=5}^{15}\frac{1}{j^2+1}}$ .

Concept Used:

In this, use the concept of putting the value of variable in the equation and adding it and get the answer and use the concept of nth term of sum of the series.

Calculation: Here, the equation is ${\displaystyle \sum _{j=5}^{15}\frac{1}{j^2+1}}$ .

  $\begin{array}{l}{\displaystyle \sum _{j=5}^{15}\frac{1}{j^2+1}}=\frac{1}{5^2+1}+\frac{1}{6^2+1}+\frac{1}{7^2+1}+\mathrm{.............}+\frac{1}{{15}^2+1}\\ {\displaystyle \sum _{j=5}^{15}\frac{1}{j^2+1}}=\frac{1}{26}+\frac{1}{37}+\frac{1}{50}+\mathrm{.............}+\frac{1}{226}\\ {\displaystyle \sum _{j=5}^{15}\frac{1}{j^2+1}}=0.038+0.027+0.02+0.015+0.012+\mathrm{........}+0.005+0.004\\ {\displaystyle \sum _{j=5}^{15}\frac{1}{j^2+1}}\approx 0.153\\ \end{array}$

Here as the number of terms increases the value of each single term decreases, so last few terms of the sum can be neglected because there are very small. Hence the value of given summation will approximately $0.153$ .

Conclusion:

The summation is $0.153$ .