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. $$\sum_{n=1}^{\infty} \frac{1}{n^{2}}$$

Short answer

The first 5 partial sums are 1, 5/4, 49/36, 205/144, 5269/3600.

Step-by-step solution

Identify the Sequence

The series given is \( \sum_{n=1}^{\infty} \frac{1}{n^2} \). The terms of the sequence \( a_n \) are defined as \( a_n = \frac{1}{n^2} \).

Calculate the First 5 Terms

Calculate the first 5 terms of \( a_n \):- For \( n = 1 \), \( a_1 = \frac{1}{1^2} = 1 \).- For \( n = 2 \), \( a_2 = \frac{1}{2^2} = \frac{1}{4} \).- For \( n = 3 \), \( a_3 = \frac{1}{3^2} = \frac{1}{9} \).- For \( n = 4 \), \( a_4 = \frac{1}{4^2} = \frac{1}{16} \).- For \( n = 5 \), \( a_5 = \frac{1}{5^2} = \frac{1}{25} \).

Calculate the First 5 Partial Sums

The partial sums \( S_n \) are calculated by summing the first \( n \) terms of the sequence.- \( S_1 = a_1 = 1 \).- \( S_2 = a_1 + a_2 = 1 + \frac{1}{4} = \frac{5}{4} \).- \( S_3 = a_1 + a_2 + a_3 = 1 + \frac{1}{4} + \frac{1}{9} = \frac{49}{36} \).- \( S_4 = a_1 + a_2 + a_3 + a_4 = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} = \frac{205}{144} \).- \( S_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} = \frac{5269}{3600} \).

Graph the Sequence and Partial Sums

Create a graph showing the first 5 terms of \( a_n \) and \( S_n \) on the same axes. Plot the points:- For \( a_n \): (1, 1), (2, 0.25), (3, 0.111), (4, 0.0625), (5, 0.04).- For \( S_n \): (1, 1), (2, 1.25), (3, 1.3611), (4, 1.4236), (5, 1.4636).Label the axes and plot accordingly.

Key concepts

Partial Sums

When dealing with convergent series, understanding partial sums is crucial. A partial sum, like the ones denoted by \( S_n \) in our exercise, represents the sum of the first \( n \) terms of a sequence.
For example, to find the partial sum \( S_5 \) of the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), you add up the first five terms:

• \( S_1 = a_1 = 1 \)
• \( S_2 = a_1 + a_2 = 1 + \frac{1}{4} = \frac{5}{4} \)
• \( S_3 = a_1 + a_2 + a_3 = 1 + \frac{1}{4} + \frac{1}{9} = \frac{49}{36} \)
• \( S_4 = a_1 + a_2 + a_3 + a_4 = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} = \frac{205}{144} \)
• \( S_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} = \frac{5269}{3600} \)

Partial sums help us understand how a series behaves. As \( n \) increases, if these sums approach a particular limit, the series is called convergent.

Sequence of Terms

The term "sequence of terms" refers to the individual terms \( a_n \) of a series. In our given series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), each term \( a_n \) is calculated as \( \frac{1}{n^2} \).
For the first five terms, we have:

• \( a_1 = \frac{1}{1^2} = 1 \)
• \( a_2 = \frac{1}{2^2} = \frac{1}{4} \)
• \( a_3 = \frac{1}{3^2} = \frac{1}{9} \)
• \( a_4 = \frac{1}{4^2} = \frac{1}{16} \)
• \( a_5 = \frac{1}{5^2} = \frac{1}{25} \)

Understanding each term independently is important because they provide insight into how the sequence behaves as \( n \) increases. As is evident, each term gets smaller, reflecting a common pattern in convergent series.

Graphing Sequences

Graphing sequences helps visualize the behavior of sequences and the progression of their partial sums. By plotting each term \( a_n \) and the partial sums \( S_n \), we can see the relationship between them.
In our example, we plotted:

• Terms \( a_n \): (1, 1), (2, 0.25), (3, 0.111), (4, 0.0625), (5, 0.04)
• Partial Sums \( S_n \): (1, 1), (2, 1.25), (3, 1.3611), (4, 1.4236), (5, 1.4636)

With these points graphed on the same axes, you notice how \( a_n \) decreases rapidly while \( S_n \) gradually approaches a limit. Graphs make it easier to identify trends and convergence within a sequence and overall series.

Infinite Series

An infinite series is formed by summing the terms of a sequence endlessly. When we speak of the convergence of an infinite series, like \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), we're considering whether those sums approach a specific value as more terms are added.
An infinite series may converge or diverge. In our series, shown through the calculation of partial sums such as \( S_5 = \frac{5269}{3600} \), as more terms are added, these sums approach a particular number (in this case, a known mathematical constant, \( \pi^2/6 \)).

• A convergent series reaches a finite limit as \( n \to \infty \).
• Divergent series either grow indefinitely or oscillate without settling to a single value.

Understanding infinite series is pivotal for advanced mathematical concepts like calculus and real analysis.