Question

Calculating sums: Determine the value of each of the sums that

follow. Some can be computed directly, some require the use

of sum formulas, and for some you will have to also compute

a limit .

$\underset{k=1}{\overset{10}{\sum \frac{1}{k^2}}}$

Short answer

The value is$\frac{1968329}{1270080}$.

Step-by-step solution

Step 1. Given information .

Consider the given sigma function$\underset{k=1}{\overset{10}{\sum \frac{1}{k^2}}}$.

Step 2. Find the value .

Compute $a=1,2,3...............10$in the given sigma function .

When k = 1 then role="math" localid="1648722671346" $\frac{1}{k^2}=\frac{1}{1^{{}^2}}=1=a_1$

k = 2 then $a_2=\frac{1}{4}$

k = 3 then $a_3=\frac{1}{9}$

k = 4 then $a_4=\frac{1}{16}$

k= 5 then $a_5=\frac{1}{25}$

k= 6 then $a_6=\frac{1}{36}$

k = 7 then $a_7=\frac{1}{49}$

k= 8 then $a_8=\frac{1}{64}$

k= 9 then $a_9=\frac{1}{81}$

k =10 then $a_{10}=\frac{1}{100}$

Compute the sum of $a_k=\frac{1}{k^2}=1.............10$.

$$\begin{gathered} =1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{49}+\frac{1}{64}+\frac{1}{81}+\frac{1}{100} \\ =\frac{1968329}{1270080} \end{gathered}$$