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. $$\sum_{n=1}^{\infty}\left(-\frac{9}{10}\right)^{n}$$

Short answer

The first five partial sums are -0.9, -0.09, -0.819, -0.1629, and -0.75339.

Step-by-step solution

Identify the Terms

The series given is \( \sum_{n=1}^{\infty} \left(-\frac{9}{10}\right)^{n} \). This is a geometric series where \( a_1 = -\frac{9}{10} \) and the common ratio \( r = -\frac{9}{10} \).

Calculate the First 5 Terms

For a geometric sequence \( a_n = a_1 \cdot r^{n-1} \). The first term is \( a_1 = -\frac{9}{10} \), the second term is \( a_2 = \left(-\frac{9}{10}\right)^2 = \frac{81}{100} \), the third term is \( a_3 = \left(-\frac{9}{10}\right)^3 = -\frac{729}{1000} \), the fourth term is \( a_4 = \left(-\frac{9}{10}\right)^4 = \frac{6561}{10000} \), and the fifth term is \( a_5 = \left(-\frac{9}{10}\right)^5 = -\frac{59049}{100000} \).

Calculate the Partial Sums

The nth partial sum \( S_n \) of a geometric series is given by \( S_n = a_1 \frac{1-r^n}{1-r} \). Calculate: \( S_1 = -0.9 \), \( S_2 = -0.9 + 0.81 = -0.09 \), \( S_3 = -0.9 + 0.81 - 0.729 = -0.09 - 0.729 = -0.819 \), \( S_4 = -0.819 + 0.6561 = -0.1629 \), \( S_5 = -0.1629 - 0.59049 = -0.75339 \).

Sketch a Graph

On a single set of axes, plot the 5 terms of the sequence \( a_n \) and the partial sums \( S_n \). The x-axis represents the term number \( n \). For \( a_n \), plot: \( (-0.9, 0.81, -0.729, 0.6561, -0.59049) \), and for \( S_n \), plot: \( (-0.9, -0.09, -0.819, -0.1629, -0.75339) \). This visualizes the alternating nature of the terms and how the sum converges.

Key concepts

Partial Sums

A partial sum is like taking a snapshot of how much you've added up at a certain point in a series. Imagine a never-ending series as a long journey. Partial sums help you see how far you've gone after taking a few steps. In mathematical terms, partial sums are the accumulated total of the first few terms in an infinite series.
For geometric series, calculating partial sums involves using the formula:

• \( S_n = a_1 \frac{1-r^n}{1-r} \)

Where:

• \( S_n \) is the nth partial sum
• \( a_1 \) is the first term
• \( r \) is the common ratio

Each time you add another term to your partial sum, you update your total. In the example provided, the partial sums illustrate the changes each added term makes to the overall sum, showing a zig-zag pattern as the terms alternate in sign due to a negative ratio.

Convergence

When discussing series, convergence is like discovering where the series is heading. A series converges if its partial sums get closer and closer to a specific value as more terms are added. If this value exists, the series is convergent; otherwise, it's divergent.
For a geometric series:

• The series converges when the absolute value of the common ratio \( r \) is less than 1: \( |r| < 1 \)

In our example, the common ratio \( r = -\frac{9}{10} \), and indeed \(|r| = 0.9 < 1\) which implies convergence. This means that as you add more terms from the series, the changes become smaller, and the sum gets closer to a point where it ‘settles’. It's like a train that finally stops after adjusting its speed over time.

Geometric Sequence

A geometric sequence is a list of numbers where each term is a fixed multiple of the previous one. This multiplicative factor is known as the common ratio, \( r \). For example, in a geometric sequence \( a_1, a_2, a_3, \ldots \), each term is given by:

• \( a_n = a_1 \cdot r^{n-1} \)

Here, \( a_1 \) is the first term, and \( r \) is consistent for every step. This creates a pattern of growth or decay.
In the exercise provided, the negative common ratio \( r = -\frac{9}{10} \) results in the sequence alternating between negative and positive values—demonstrating how geometric sequences can vary greatly depending on the ratio. They can quickly grow large or shrink small, revealing interesting behaviors in sequences and series!