Answer
$\dfrac{25}{12}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To evaluate the given summation expression, $
\displaystyle\sum_{j=1}^4 j^{-1}
,$ substitute $
j
$ with the values from $
1
$ to $
4
$ and then simplify the expression.
$\bf{\text{Solution Details:}}$
Substituting $
j
$ with the numbers from $
1
$ to $
4
,$ the given expression evaluates to
\begin{array}{l}\require{cancel}
1^{-1}+2^{-1}+3^{-1}+4^{-1}
.\end{array}
Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{1^1}+\dfrac{1}{2^1}+\dfrac{1}{3^1}+\dfrac{1}{4^1}
\\\\=
\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}
\\\\=
\dfrac{12}{12}+\dfrac{6}{12}+\dfrac{4}{12}+\dfrac{3}{12}
\\\\=
\dfrac{25}{12}
.\end{array}
