Question

How do you find the sum of $\sum {\dfrac{1}{{{k^2} + 1}}}$ where $k$ is $\left[ {0,3} \right]$ ?

Answer 1

In this question, we are given an expression involving sigma and we have to find the sum within the given range of $k$. Put all the values of $k$ one by one and then, add all the terms. Note the brackets used to show the range. See whether they are open brackets or close brackets. Accordingly, put all the values of $k$.

Complete step-by-step solution:
We are given an expression involving the variable $k$ and we are also given the potential values of $k$. Let us put the values of $k$.
$\Rightarrow \sum {\dfrac{1}{{{k^2} + 1}}}$ ………….. (given)
The values that we have to put are $0$ , $1$ , $2$ and $3$ .
Let us put the values one by one and then add each of the terms.
$\Rightarrow \dfrac{1}{{{0^2} + 1}} + \dfrac{1}{{{1^2} + 1}} + \dfrac{1}{{{2^2} + 1}} + \dfrac{1}{{{3^2} + 1}}$
Now, we will simplify the denominator of all the terms.
$\Rightarrow \dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{{4 + 1}} + \dfrac{1}{{9 + 1}}$
Simplifying them further,
$\Rightarrow \dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{5} + \dfrac{1}{{10}}$
Now, notice that all the terms have different denominators. In order to add the terms, we need the denominator to be the same. We will take the LCM.
LCM of $1$, $2$, $5$ and $10$ = $10$
Converting the denominators into $10$ ,
$\Rightarrow \dfrac{{10}}{{10}} + \dfrac{5}{{10}} + \dfrac{2}{{10}} + \dfrac{1}{{10}}$
Adding all the terms,
$\Rightarrow \dfrac{{10 + 5 + 2 + 1}}{{10}} = \dfrac{{18}}{{10}} = 1.8$

Hence, $\sum {\dfrac{1}{{{k^2} + 1}}} = 1.8$.

Note: Open and Close brackets:
There are two types of brackets,
i) Open Brackets: The round brackets are called open brackets - $\left( - \right)$. If these brackets are used, then the numbers are not included. For example: $\left( {0,3} \right)$ - this includes the digits $1,2$ only.
ii) Close brackets: The square brackets are called close brackets - $\left[ - \right]$. If these brackets are used, then the numbers are included. For example: $\left[ {0,3} \right]$ - this includes the digits $0$ , $1$ , $2$ and $3$ .
Sometimes, a combination of these brackets is also used - $( - ]$ or $[ - )$ .
For example: $(0,3]$ includes numbers $1,2,3$.
Similarly, $[0,3)$ includes numbers $0,1,2$ .