Question

Let \(\sum a_{n}\) be the series whose \(n\) th term is \(a_{n}=\frac{1}{2^{n}}-\frac{1}{3^{n}}, n \geq 1\). Find the first ten terms, the first ten partial sums, and plot the sequence and partial sums. Do you think the series converges? If so, what is the sum?

Short answer

Yes, the series converges to \( \frac{1}{2} \).

Step-by-step solution

Identify the first ten terms

The given series is \( a_{n} = \frac{1}{2^n} - \frac{1}{3^n} \). Calculate the terms for \( n = 1 \) to \( n = 10 \).\- For \( n = 1, a_{1} = \frac{1}{2^1} - \frac{1}{3^1} = \frac{1}{2} - \frac{1}{3} = \frac{1}{6} \).\- For \( n = 2, a_{2} = \frac{1}{2^2} - \frac{1}{3^2} = \frac{1}{4} - \frac{1}{9} = \frac{5}{36} \).\Continue computing similarly up to \( n = 10 \).

Calculate the first ten partial sums

Sum the first \( n \) terms of the sequence computed in Step 1 to find the partial sums. Let \( S_n \) represent the partial sum of the first \( n \) terms of \( a_n \).\- \( S_1 = a_1 = \frac{1}{6} \).\- \( S_2 = a_1 + a_2 = \frac{1}{6} + \frac{5}{36} = \frac{11}{36} \).\Continue computing similarly up to \( S_{10} \).

Plot the sequence and partial sums

Using the terms \( a_n \) and the partial sums \( S_n \) computed in Steps 1 and 2, plot the sequence \( a_n \) against \( n \) and the partial sums \( S_n \) against \( n \). This visual helps observe the behavior of the series and partial sums for the first ten terms.

Analyze convergence

Since \( a_n = \frac{1}{2^n} - \frac{1}{3^n} \) is a difference of two geometric series, both with common ratio less than 1, their differences as \( n \rightarrow \infty \) will approach 0. The partial sums for this series resemble a converging geometric series. Therefore, the series converges.

Find the sum of the series

Each series \( \sum \frac{1}{2^n} \) and \( \sum \frac{1}{3^n} \) is a geometric series with sums \( \frac{1}{1-r} \), where \( r \) is the ratio.\- The sum \( \sum \frac{1}{2^n} = \frac{1/2}{1-1/2} = 1 \).\- The sum \( \sum \frac{1}{3^n} = \frac{1/3}{1-1/3} = \frac{1}{2} \).\- The difference \( \sum a_n = 1 - \frac{1}{2} = \frac{1}{2} \). Thus, the sum of the series is \( \frac{1}{2} \).

Key concepts

Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant called the common ratio. In our case, the sequence \( a_n = \frac{1}{2^n} - \frac{1}{3^n} \) includes two geometric series. The first series, \( \sum \frac{1}{2^n} \), has a common ratio of \( \frac{1}{2} \), while the second series, \( \sum \frac{1}{3^n} \), uses \( \frac{1}{3} \) as its ratio.

Geometric series are significant for their predictable behavior that allows for straightforward calculation of sums. A geometric series converges if its common ratio's absolute value is less than one. The sum of an infinite geometric series can be found using the formula: \[ S = \frac{a}{1 - r} \]where \( a \) is the first term and \( r \) is the common ratio.

In our example, each geometric series converges separately, leading to the convergence of their difference, which is the given series.

Partial Sums

Partial sums are a method for examining an infinite series by summing a finite number of its terms. They help us understand how the series behaves as more terms are added. For the series \( a_n = \frac{1}{2^n} - \frac{1}{3^n} \), each partial sum \( S_n \) is calculated by adding up the first \( n \) terms of the series.

For example, the first partial sum \( S_1 \) is simply \( a_1 = \frac{1}{6} \). The second partial sum \( S_2 \) is \( \frac{1}{6} + \frac{5}{36} = \frac{11}{36} \). This process is repeated, adding the next term to the existing sum.

Partial sums are crucial for identifying whether a series converges. If their values tend towards a fixed number as \( n \) increases, the series is said to be convergent. In our series, the partial sums get closer to \( \frac{1}{2} \), suggesting convergence.

Sequence Plotting

Plotting sequences and partial sums provides a visual representation of how terms are progressing as \( n \) changes. For the series \( a_n = \frac{1}{2^n} - \frac{1}{3^n} \), you can plot individual terms \( a_n \) against their position \( n \) to track the decay towards zero.

Similarly, plotting the partial sums \( S_n \) against \( n \) allows one to see the trend toward a specific value, which in this case appears to stabilize around \( \frac{1}{2} \) as more terms are added.

Sequence plotting is beneficial for visual learners, providing a graphic means to assess whether a series converges or diverges. It illustrates the technical calculation steps graphically, helping deepen the understanding.

Mathematical Analysis

Mathematical analysis involves rigorously examining sequences and series to determine characteristics like convergence. For the sequence defined by \( a_n = \frac{1}{2^n} - \frac{1}{3^n} \), mathematical analysis reveals it as the difference of two convergent geometric series.

As \( n \) approaches infinity, each term of the sequence approaches zero because the geometric series involved have ratios less than one. Analyzing the behavior of terms allows conclusion about the whole series. Here, rigorous analysis helps establish that the series converges to \( \frac{1}{2} \), since the limits of the series' component sums are 1 and \( \frac{1}{2} \) respectively.

Mathematical analysis not only helps in determining convergence but also in calculating the exact sum, providing a deep foundation for understanding series.