Question

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. (a) Give the first 5 partial sums of the series. (b) Give a graph of the first 5 terms of \(a_{n}\) and \(S_{n}\) on the same axes. $$\sum_{n=1}^{\infty} n$$

Short answer

The first 5 partial sums are 1, 3, 6, 10, 15. Plot both nth terms and partial sums.

Step-by-step solution

Identify the Terms

The series in question is \( \sum_{n=1}^{\infty} n \), which means that each term of the series is simply \( n \). Thus, the first few terms are 1, 2, 3, 4, 5, etc.

Calculate Partial Sums

To find the partial sums, we add the first few terms of the series together. 1. First Partial Sum (\( S_1 \)) = 1 2. Second Partial Sum (\( S_2 \)) = 1 + 2 = 3 3. Third Partial Sum (\( S_3 \)) = 1 + 2 + 3 = 6 4. Fourth Partial Sum (\( S_4 \)) = 1 + 2 + 3 + 4 = 10 5. Fifth Partial Sum (\( S_5 \)) = 1 + 2 + 3 + 4 + 5 = 15.

Graph the Terms and Partial Sums

On a graph with the x-axis representing the index \( n \) and the y-axis representing the value, plot the points for both the individual terms \( a_n = n \) and the partial sums \( S_n \). For \( n = 1, 2, 3, 4, 5 \), plot the points \((1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\) for \( a_n \) and \((1, 1), (2, 3), (3, 6), (4, 10), (5, 15)\) for \( S_n \). Draw lines connecting these points to help visualize the progression.

Key concepts

Partial Sums

When we talk about partial sums, we're essentially adding up the beginning of an infinite series to understand its progression. Take it step by step: a partial sum is the sum of the first few terms of a series. For instance, in the series \( \sum_{n=1}^{\infty} n \), each term is simply \( n \). Starting with the first term, each subsequent partial sum includes one more term from the series.
The concept of partial sums is crucial because it helps us understand how the series behaves as we add more terms:

• The first partial sum \( S_1 = 1 \)
• The second partial sum \( S_2 = 1 + 2 = 3 \)
• The third partial sum \( S_3 = 1 + 2 + 3 = 6 \)
• The fourth partial sum \( S_4 = 1 + 2 + 3 + 4 = 10 \)
• The fifth partial sum \( S_5 = 1 + 2 + 3 + 4 + 5 = 15 \)

Notice how the values increase as you add more terms. This reflects the nature of convergence or divergence in infinite series, which can be further explored through the partial sums.

Graphing Sequences

Graphing is a powerful way to visualize sequences and series. Graphing sequences involves plotting points on a graph where the x-axis represents the index \( n \), and the y-axis represents the value of the sequence or series at that index. For the series \( \sum_{n=1}^{\infty} n \), the terms are simply \( n \), which you can think of as a simple counting sequence.
Graphing the sequence can help us see patterns or trends, such as whether the terms are increasing or decreasing, and how quickly. Here's how you do it:

• For \( a_n = n \), plot points like \((1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\).
• For partial sums \( S_n \), plot \((1, 1), (2, 3), (3, 6), (4, 10), (5, 15)\).

Drawing lines between points can help us visualize how the series grows over time. This visual can make it easier to grasp the idea of growth, showing how fast or slow a series might approach infinity.

Terms of a Series

A series is essentially the sum of the terms of a sequence. Each term in a series is a fundamental part of the sequence, and understanding these terms is critical in analyzing the series itself. In the series \( \sum_{n=1}^{\infty} n \), each term is simply the index \( n \), which means each term increases by 1.

• The first term \( a_1 = 1 \)
• The second term \( a_2 = 2 \)
• The third term \( a_3 = 3 \)
• The fourth term \( a_4 = 4 \)
• The fifth term \( a_5 = 5 \)

Identifying each term allows us to delve into more complex questions like determining convergence or divergence and the behavior of the series. When we deal with infinite series, the nature of these terms heavily influences whether the series can be summed to a finite value, or if it will grow without bound. Seeing a series as a sum of its parts—its terms—can make the broader concept much clearer.