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. $$\sum_{n=1}^{\infty} \frac{1}{3^{n}}$$

Short answer

The first 5 partial sums are \( \frac{1}{3}, \frac{4}{9}, \frac{13}{27}, \frac{40}{81}, \frac{121}{243} \).

Step-by-step solution

Define the terms of the series

In the given series, \( a_{n} = \frac{1}{3^{n}} \). This defines each term of the series as the reciprocal of 3 raised to the power of \( n \).

Compute the first 5 terms

Calculate the first five terms of the series: \( a_{1} = \frac{1}{3^{1}} = \frac{1}{3} \), \( a_{2} = \frac{1}{3^{2}} = \frac{1}{9} \), \( a_{3} = \frac{1}{3^{3}} = \frac{1}{27} \), \( a_{4} = \frac{1}{3^{4}} = \frac{1}{81} \), \( a_{5} = \frac{1}{3^{5}} = \frac{1}{243} \).

Calculate the first 5 partial sums

Calculate the partial sums by adding each successive term: \( S_{1} = \frac{1}{3} \), \( S_{2} = \frac{1}{3} + \frac{1}{9} = \frac{4}{9} \), \( S_{3} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} = \frac{13}{27} \), \( S_{4} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} = \frac{40}{81} \), \( S_{5} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} = \frac{121}{243} \).

Plot the graph

Plot the values of \( a_{n} \) and \( S_{n} \) on the same graph with the x-axis representing the term number \( n \) and the y-axis showing the value. For \( n = 1 \) to \( n = 5 \), plot \( a_{n} = \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243} \) and \( S_{n} = \frac{1}{3}, \frac{4}{9}, \frac{13}{27}, \frac{40}{81}, \frac{121}{243} \). This helps visualize both the individual terms and the cumulative sums.

Key concepts

Partial Sums

Partial sums are the sums of the first few terms of a series. They serve as a building block for understanding how the series progresses. Each partial sum includes all the terms from the initial term up to the current term. In this exercise, we consider the series \( \sum_{n=1}^{\infty} \frac{1}{3^n} \). Here, a partial sum \( S_n \) is the sum of the series' first \( n \) terms. For example:

• \( S_1 = \frac{1}{3} \)
• \( S_2 = \frac{1}{3} + \frac{1}{9} = \frac{4}{9} \)
• \( S_3 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} = \frac{13}{27} \)
• \( S_4 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} = \frac{40}{81} \)
• \( S_5 = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} = \frac{121}{243} \)

Partial sums help evaluate the behavior of a series over time. They are essential in determining whether a series converges or diverges.

Convergence of Series

Series convergence is a critical concept in calculus, referring to whether a series approaches a certain value as you add more terms. In this series \( \sum_{n=1}^{\infty} \frac{1}{3^n} \), it is important to recognize that it is a geometric series. A geometric series has a common ratio between consecutive terms, here \( \frac{1}{3} \).A geometric series converges if the absolute value of the common ratio is less than one. Thus, because \( \left| \frac{1}{3} \right| < 1 \), the series \( \sum_{n=1}^{\infty} \frac{1}{3^n} \) converges. When a series converges, it approaches a limiting value, known as the sum of the series, which can be calculated using the formula for the sum of an infinite geometric series:\[S = \frac{a}{1-r}\]For our series, the first term \( a = \frac{1}{3} \) and the common ratio \( r = \frac{1}{3} \). Thus, the series converges to:\[S = \frac{\frac{1}{3}}{1-\frac{1}{3}} = \frac{1}{2}\]This demonstrates that the series approaches the value \( \frac{1}{2} \) as the number of terms increases indefinitely.

Graphical Representation of Series

Visualizing the terms and sums of a series through graphs is helpful for understanding its behavior over time. For this series, you can plot both \( a_n \) and the partial sums \( S_n \) on the same graph.To create the graph:

• Plot the term values \( a_n \): \( \frac{1}{3} \), \( \frac{1}{9} \), \( \frac{1}{27} \), \( \frac{1}{81} \), \( \frac{1}{243} \) on the y-axis, against their corresponding term numbers \( n \) on the x-axis.
• Also, plot the partial sums \( S_n \): \( \frac{1}{3} \), \( \frac{4}{9} \), \( \frac{13}{27} \), \( \frac{40}{81} \), \( \frac{121}{243} \).

The graph provides a visual display of how quickly the series terms decrease since they become smaller as \( n \) increases, and how the partial sums approach the convergence limit (\( \frac{1}{2} \)).Graphing gives a clear, visual means to understand the nature of series convergence, demonstrating how partial sums stabilize as more terms are added.