The given sum involves square roots in the numerator and powers in the denominator. Observing the pattern, the sum can be written using sigma notation as follows:

$\sum_{k=1}^n \frac{\sqrt{k}}{k^9}$

Simplifying the denominator, $k^9$, into a single term:

$\sum_{k=1}^n \frac{\sqrt{k}}{k^9} = \sum_{k=1}^n k^{-\frac{17}{2}}$

This is the sigma notation representation for the given sum.

Would you like further details or a step-by-step explanation?
Here are 5 related questions to explore:

1. How do we generalize this sum for $n$ tending to infinity?
2. What is the difference between geometric and arithmetic series in summation problems?
3. How do exponents and roots combine in fractional powers?
4. How does this sum change if we replace $\sqrt{k}$ with another function, like $\ln(k)$?
5. What are common applications of sigma notation in physics and engineering?

Tip: When working with sigma notation, always check the pattern carefully to identify the general term correctly.