Question

Express each sum using summation notation. \(3+\frac{3^{2}}{2}+\frac{3^{3}}{3}+\cdots+\frac{3^{n}}{n}\)

Short answer

\(\sum_{k=1}^{n} \frac{3^k}{k}\)

Step-by-step solution

- Identify the general term

Observe the pattern in the sum: the general term appears to be \(\frac{3^k}{k}\) for the k-th term in the sequence.

- Determine the range of the index

Identify the starting and ending values of the index. The series starts from \(k = 1\) and goes up to \(k = n\).

- Write the summation notation

Combine the general term and the range to write the entire sum using summation notation. Thus, the sum can be written as: \(\sum_{k=1}^{n} \frac{3^k}{k}\).

Key concepts

Series

A series is the sum of the terms of a sequence. When you see a list of numbers being added together, you are looking at a series. In the given exercise, the series is the sum of terms that follow a specific pattern: \(3 + \frac{3^{2}}{2} + \frac{3^{3}}{3} + \cdots + \frac{3^{n}}{n}\).
One important thing to note about series is that they can be finite or infinite. For instance, the example given is a finite series that ends at \(\frac{3^{n}}{n}\).
In contrast, an infinite series continues forever. It's also essential to differentiate between an arithmetic series, where each term is a constant difference from the previous one, and a geometric series, where each term is multiplied by a constant factor.
To express a series concisely and to see the sum more clearly, we use notation like summation.

General Term

The general term in a series indicates the pattern or rule followed by the sequence of terms. For the series given in the exercise, the general term is identified by observing the patterns in each term.
Each term in the sequence has the form \(\frac{3^k}{k}\), where \(k\) represents the position of the term. This formula allows us to calculate any term in the series based on its position.
The general term is crucial because it helps us write and understand the series in a compact way, especially when using summation notation.

Index Range

The index range of a series specifies the starting and ending values for the index variable, indicating which terms should be included in the sum.
In the exercise, the index \(k\) starts at 1 and goes up to \(n\). This means that the series includes all terms where \(k = 1, 2, 3, \ldots, n\).
Identifying the index range is essential for understanding the scope of the series and ensuring that all relevant terms are included.
Proper definition of the index range helps in correctly writing the series in summation notation. Missing or incorrectly defining the index range may lead to an incomplete or incorrect sum.

Notation

Summation notation, also known as sigma notation, is a convenient way to write the sum of a sequence of terms. In the exercise solution, we use summation notation to write the series concisely.
The sigma (∑) symbol represents summation, and below it, a variable (like \(k\)) indicates the starting value, while above it indicates the ending value.
For example, \(\sum_{k=1}^{n} \frac{3^k}{k}\) means 'sum from \(k = 1\) to \(k = n\)' of the term \(\frac{3^k}{k}\).
Simplifying sums using summation notation allows us to handle more complex series more easily. It is a powerful tool in mathematics for expressing lengthy sums clearly and efficiently.