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. $$\sum_{n=1}^{\infty}\left(\frac{1}{10}\right)^{n}$$

Short answer

Partial sums are \( \frac{1}{10}, \frac{11}{100}, \frac{111}{1000}, \frac{1111}{10000}, \frac{11111}{100000} \).

Step-by-step solution

Identify General Term

The series is given as \( \sum_{n=1}^{\infty} \left(\frac{1}{10}\right)^{n} \). The general term of this geometric series is \( a_n = \left(\frac{1}{10}\right)^n \).

Compute First 5 Terms

Calculate the first five terms \( a_1 \) to \( a_5 \) of the series:- \( a_1 = \left(\frac{1}{10}\right)^1 = \frac{1}{10} \)- \( a_2 = \left(\frac{1}{10}\right)^2 = \frac{1}{100} \)- \( a_3 = \left(\frac{1}{10}\right)^3 = \frac{1}{1000} \)- \( a_4 = \left(\frac{1}{10}\right)^4 = \frac{1}{10000} \)- \( a_5 = \left(\frac{1}{10}\right)^5 = \frac{1}{100000} \)

Calculate First 5 Partial Sums

Calculate the first five partial sums \( S_1 \) to \( S_5 \):- \( S_1 = a_1 = \frac{1}{10} \)- \( S_2 = a_1 + a_2 = \frac{1}{10} + \frac{1}{100} = \frac{11}{100} \)- \( S_3 = a_1 + a_2 + a_3 = \frac{11}{100} + \frac{1}{1000} = \frac{111}{1000} \)- \( S_4 = a_1 + a_2 + a_3 + a_4 = \frac{111}{1000} + \frac{1}{10000} = \frac{1111}{10000} \)- \( S_5 = a_1 + a_2 + a_3 + a_4 + a_5 = \frac{1111}{10000} + \frac{1}{100000} = \frac{11111}{100000} \)

Prepare Graph Data

List data points for both the terms and partial sums for graphing:- Terms \( a_n \): \( (1, \frac{1}{10}), (2, \frac{1}{100}), (3, \frac{1}{1000}), (4, \frac{1}{10000}), (5, \frac{1}{100000}) \)- Partial sums \( S_n \): \( (1, \frac{1}{10}), (2, \frac{11}{100}), (3, \frac{111}{1000}), (4, \frac{1111}{10000}), (5, \frac{11111}{100000}) \)

Plot the Graph

Using a graphing tool, plot the values of \( a_n \) and \( S_n \) on the same axes. Label the x-axis as the term index and the y-axis for values. Plot the points for \( a_n \) and \( S_n \) with distinct markers or lines.

Key concepts

Partial Sums

In the context of geometric series, a partial sum is the sum of the first few terms of that series. For the given series \(\sum_{n=1}^{\infty}\left(\frac{1}{10}\right)^{n}\), the task is to calculate the first five partial sums. To obtain a partial sum, you add the terms sequentially. Let's walk through the process of finding the partial sums:\[S_1 = a_1 = \frac{1}{10}\] This is simply the first term itself.
To find \(S_2\), you add the second term to \(S_1\):

• \[S_2 = S_1 + a_2 = \frac{1}{10} + \frac{1}{100} = \frac{11}{100}\]

Continuing this pattern helps build a clearer picture of how sums accumulate. For example, with \(S_3\), you add \(a_3\) to \(S_2\). Similarly, for \(S_4\) and \(S_5\), each subsequent partial sum is the total of all preceding terms added together with the current term. It's about progressively gathering all terms:

• \[S_3 = S_2 + a_3 = \frac{11}{100} + \frac{1}{1000} = \frac{111}{1000}\]
• \[S_4 = S_3 + a_4 = \frac{111}{1000} + \frac{1}{10000} = \frac{1111}{10000}\]
• \[S_5 = S_4 + a_5 = \frac{1111}{10000} + \frac{1}{100000} = \frac{11111}{100000}\]

Observing the growth of partial sums is essential in understanding how series behave, especially as they converge towards a certain number.

Graph of a Series

Visual representation of a series provides an insightful dimension to its comprehension. For our series \(\sum_{n=1}^{\infty}\left(\frac{1}{10}\right)^{n}\), plotting both the terms \(a_n\) and the partial sums \(S_n\) on the same graph enhances understanding.
We start by listing the terms and partial sums:

• Terms \(a_n\): \((1, \frac{1}{10}), (2, \frac{1}{100}), (3, \frac{1}{1000}), (4, \frac{1}{10000}), (5, \frac{1}{100000})\)
• Partial sums \(S_n\): \((1, \frac{1}{10}), (2, \frac{11}{100}), (3, \frac{111}{1000}), (4, \frac{1111}{10000}), (5, \frac{11111}{100000})\)

To draw the graph:

• Label the x-axis as the term index (1 to 5) and the y-axis for values.
• Plot the points for the terms and join them with lines for a clearer trend.
• Use distinct markers for partial sums \(S_n\) and terms \(a_n\) to differentiate them easily.

This graph helps in visually connecting how each term contributes to the partial sum. It shows how the partial sums gradually level off or stabilize, illustrating the concept of convergence graphically.

Convergence of Series

Convergence is a vital concept in studying series, especially for geometric series. A series like \(\sum_{n=1}^{\infty}\left(\frac{1}{10}\right)^{n}\) converges if its partial sums approach a specific limit as \(n\) becomes large.
For a geometric series of the form \(\sum_{n=0}^{\infty} ar^n\), convergence is determined by the common ratio \(r\).

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), it diverges.

In our case, \(r = \frac{1}{10}\), which is less than 1, confirming that the series converges. The series converges to the sum \(S = \frac{a}{1 - r}\), where \(a\) is the first term.
For \(a = \frac{1}{10}\) and \(r = \frac{1}{10}\), the sum of the series is:\[S = \frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{1}{9}\]This indicates that as more terms are added, the partial sums get closer to \(\frac{1}{9}\), even though they never actually reach it. Understanding convergence helps predict long-term behavior of series and is fundamental in various mathematical applications.