What is the exact value of the sum from k=2 to 4 of (k^(-2) * (1/k))?

To find the sum from ( k=2 ) to ( 4 ) of ( k^{-2} \cdot \frac{1}{k} ), we first simplify the expression. The term can be rewritten as:

[

k^{-2} \cdot \frac{1}{k} = \frac{1}{k^2} \cdot \frac{1}{k} = \frac{1}{k^3}

]

Now, we evaluate the sum ( \sum_{k=2}^{4} \frac{1}{k^3} ):

• For ( k=2 ):

[

\frac{1}{2^3} = \frac{1}{8}

]

• For ( k=3 ):

[

\frac{1}{3^3} = \frac{1}{27}

]

• For ( k=4 ):

[

\frac{1}{4^3} = \frac{1}{64}

]

Now, we combine these fractions to get the total sum:

[

\frac{1}{8} + \frac{1}{27} + \