Q. 1 - Calculus

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 .

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

Step-by-Step Solution

Answer

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

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 $\frac{1}{k^2}=\frac{1}{1^{{}^2}}=1=a_1$

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 $a_k=\frac{1}{k^2}=1.............10$.

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