To express the given series $3 + \frac{9}{2} + \frac{27}{3} + \frac{81}{4} + \cdots + \frac{59,049}{10}$ in summation notation, we need to analyze its pattern.

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Step 1: Observe the Numerator

The numerators are: $3, 9, 27, 81, \dots, 59,049$ These terms form a geometric sequence: $3, 3^2, 3^3, 3^4, \dots, 3^{10}$

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Step 2: Observe the Denominator

The denominators are: $1, 2, 3, 4, \dots, 10$ This is simply the sequence of natural numbers $k = 1, 2, 3, \dots, 10$.

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Step 3: Generalize the $k$-th Term

For the $k$-th term of the series, the numerator is $3^k$ and the denominator is $k$. Therefore, the general term can be written as: $\frac{3^k}{k}$

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Step 4: Write in Summation Notation

The series can now be written as: $\sum_{k=1}^{10} \frac{3^k}{k}$

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Final Answer:

$\sum_{k=1}^{10} \frac{3^k}{k}$

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Relative Questions:

1. What is the general form of a geometric sequence?
2. How do we determine the numerator and denominator patterns in summation problems?
3. What are common techniques to simplify summation expressions?
4. How can this summation be computed or approximated?
5. What are other examples of summations involving powers or factorials?

Tip:

When dealing with summation problems, always break the series into two parts: look for patterns in the numerators and denominators separately.