Q. 24 - Calculus

Question

Write each of the sums in Exercises 21–28 in sigma notation. Identify $m,n,a_k$ in each problem.  

$\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{49}+\frac{1}{64}$

Step-by-Step Solution

Answer

The sigma notation is $\sum _{k=2}^8\frac{1}{k^2}and m=2,n=8,a_k=\frac{1}{k^2}$

Step 1. Given Information

The given sum is $\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{49}+\frac{1}{64}$

Step 2. Explanation

The sum in sigma notation is written as, 

$$\begin{gathered} \frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{49}+\frac{1}{64}=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+\frac{1}{8^2} \\ =\sum _{k=2}^8\frac{1}{k^2} \\ where, \\ m=2,n=8,a_k=\frac{1}{k^2} \end{gathered}$$