Question

For the following series, write formulas for the sequences \(a_{n}, S_{n},\) and \(R_{n},\) and find the limits of the sequences as \(n \rightarrow \infty\) (if the limits exist). $$\frac{3}{1 \cdot 2}-\frac{5}{2 \cdot 3}+\frac{7}{3 \cdot 4}-\frac{9}{4 \cdot 5}+\cdots$$

Short answer

Formulas: \(a_n = (-1)^{n+1} \frac{2n+1}{n(n+1)}\), \(S_n = \sum_{k=1}^{n} a_k\), \(R_n = S - S_n\). As \( n \rightarrow \infty \): \( \ a_n \rightarrow 0 \), \ \ S_n \rightarrow 0 \), \(R_n \rightarrow 0 \).

Step-by-step solution

- Identify the Sequence

Given series: \( \frac{3}{1 \times 2} - \frac{5}{2 \times 3} + \frac{7}{3 \times 4} - \frac{9}{4 \times 5} + \text{...} \)Let the general term of the sequence be \( a_n \). Identify the pattern in the numerators and denominators.

- General Term of the Sequence (\(a_{n}\))

Observe the pattern: The numerator increases by 2, starting from 3. The denominator is the product of two consecutive integers. Formulate \( a_n \) as follows:\( a_n = (-1)^{n+1} \frac{2n+1}{n(n+1)} \), where \( n = 1, 2, 3, \text{...} \)

- Recursive Sum (\(S_{n}\))

The recursive partial sum \( S_n \) is the sum of the first \(n\) terms of the series.\( S_n = \sum_{k=1}^{n}a_k \), substitute \( a_k = (-1)^{k+1} \frac{2k+1}{k(k+1)} \)

- Estimate the Limit of \(a_n\)

As \( n \rightarrow \infty \), analyze the behavior of \( a_n \).Simplify the term: \(a_n = (-1)^{n+1} \frac{2n+1}{n(n+1)} = (-1)^{n+1} \left( \frac{2n+1}{n^2+n} \right) \approx (-1)^{n+1} \left( \frac{2+\frac{1}{n}}{n+1} \right) \ \lim_{n \to \infty} a_n = 0\)

- Series Convergence Analysis (\( S_n \))

Since the partial sums \( S_n \) will be alternating and tend to zero, the series converges.As \( n \rightarrow \infty \), the limit of the partial sums of an alternating series is zero.

- Express \(R_{n}\)

The remainder of the series \( R_n \) is given by the limit difference between the infinite series and its partial sum.\(R_n = S - S_n\).Since \(S_n\) converges as \( n \to \infty \), \( R_n \to 0 \).

Key concepts

sequence limit

Understanding the limit of a sequence is crucial because it tells us what value the sequence approaches as the number of terms goes to infinity. For the given series, we identify the general term of the sequence as:
\(a_n = (-1)^{n+1} \frac{2n+1}{n(n+1)}\).
In this equation, the term \( (-1)^{n+1} \) signifies an alternating pattern in the sign of each term in the sequence.
As \( n \rightarrow \infty \), we simplify the term to observe its behavior:
\( a_n \approx (-1)^{n+1} \left( \frac{2 + \frac{1}{n}}{n + 1} \right) \). From this we can determine that:
\( \lim_{n \to \infty} a_n = 0 \).
So, the limit of the given sequence as \( n \) approaches infinity is 0.

partial sum

Partial sums help us understand how a series builds up as we add more terms. The partial sum \(S_n\) is the sum of the first \(n\) terms of the series. For our series, it is computed as:
\( S_n = \sum_{k=1}^{n}a_k \), where \( a_k = (-1)^{k+1} \frac{2k+1}{k(k+1)} \).
By adding more terms, we get closer to the cumulative value of the entire series.
For alternating series, each partial sum oscillates around the eventual sum of the series, generally getting closer with each term.

alternating series

An alternating series is one where the terms switch between positive and negative. In our example, each term \(a_n\) has a factor of \( (-1)^{n+1} \), causing the alternation in sign:
\( a_n = (-1)^{n+1} \frac{2n+1}{n(n+1)} \).
This pattern affects the partial sums, as each addition changes the sum to be slightly higher or lower than the previous partial sum. The Alternating Series Test helps determine whether such series converge. It requires the terms to:

• Eventually decrease in absolute value
• Approach zero

Both criteria are met in our series, thus determining that it converges.

remainder of a series

The remainder \(R_n\) of a series represents the part of the series that is left over after the first \(n\) terms have been summed. It is expressed as:
\(R_n = S - S_n\), where \(S\) is the total sum of the infinite series and \(S_n\) is the partial sum of the first \(n\) terms.
Since we determined that the partial sums of our series converge to 0, the remainder also becomes insignificantly small as \(n\) approaches infinity:\( R_n \to 0 \).
This indicates that what remains after summing the first \(n\) terms diminishes, verifying convergence.

general term of a series

Identifying the general term of a series is fundamental to understand how the series behaves. The given series:
\( \frac{3}{1 \times 2} - \frac{5}{2 \times 3} + \frac{7}{3 \times 4} - \frac{9}{4 \times 5} + \cdots \)
Observing the pattern, the numerator increases by 2 beginning from 3, while the denominator is the product of two consecutive integers. Thus, the general term is:
\( a_n = (-1)^{n+1} \frac{2n+1}{n(n+1)} \).
This formulation shows the alternating pattern and how the numerator and denominator change systematically.