Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\cdots\) Hint: \(a_{n}=\frac{1}{n(n+1)}\)

Step-by-Step Solution

Answer

The sum of the series converges to 1.

Step 1: Identify the General Term

Recognize that the general term of the series is given by \( a_n = \frac{1}{n(n+1)} \), where \( n \) starts from 1 and increases by 1 for each subsequent term.

Step 2: Simplify the General Term

Using the hint, express the term \( \frac{1}{n(n+1)} \) in a simplified form. This can be rewritten using partial fraction decomposition as follows:\[\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}\].

Step 3: Recognize the Telescoping Nature

Observe that in the series \( \frac{1}{n} - \frac{1}{n+1} \), most terms cancel out when summed over consecutive \( n \). Specifically, each \( -\frac{1}{n+1} \) cancels with \( \frac{1}{n+1} \) from the subsequent fraction.

Step 4: Write the Partial Sums

Write the first few partial sums to see the cancellation:\[\left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \cdots\]This reduces to: \( 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \cdots \).

Step 5: Determine the Limit as n Approaches Infinity

Note that as \( n \to \infty \), the last remaining uncanceled term approaches zero. Therefore, the series becomes:\[1 - \lim_{n \to \infty} \frac{1}{n+1} = 1\].

Step 6: Conclusion

Since all other terms cancel out in a telescoping manner, the sum of the infinite series converges to 1.

Key Concepts

Partial fraction decomposition explained

Partial fraction decomposition is a valuable tool in mathematics, especially when dealing with complex fractions that can be broken down into simpler parts. This technique involves expressing a rational expression as the sum of simpler fractions, which can simplify the process of evaluating series or solving integrals.
To perform partial fraction decomposition, particularly on expressions like \( \frac{1}{n(n+1)} \), you express it as a sum of two fractions: \( \frac{1}{n} - \frac{1}{n+1} \). This transformation is crucial because it enables the cancellation of terms in an infinite series, making it easier to evaluate the series.

• Start by expressing the complex fraction in terms of simpler components.
• Use algebraic manipulation to break down the original fraction.
• Apply this technique when the denominator is a product of linear factors, preparing it for telescoping series evaluations.

By breaking the fraction into smaller parts, you allow terms to cancel later in the process of summing up series, simplifying calculations significantly.

Understanding infinite series

An infinite series is a sequence of numbers, added together, that continues indefinitely. It represents the sum of an infinite number of terms. This concept is crucial in calculus and advanced mathematics, helping to evaluate sums that do not seem to end.
In the problem given, the series \( \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \cdots \) illustrates an infinite series starting from 1. Infinite series can be classified based on their convergence or divergence, which determines whether they add up to a finite number or diverge to infinity.

• Series are often defined by a general term: here it is \( \frac{1}{n(n+1)} \).
• Partial fraction decomposition helps convert this general term to simpler fractions.
• Infinite series can be used in various applications, including approximations and solving differential equations.

Understanding infinite series is fundamental in identifying how partial sums lead to convergence, which is crucial for evaluating the behavior of series in real-world applications.

Exploring sequence convergence

Sequence convergence is a key concept, especially when considering infinite series. A sequence converges when its terms approach a specific value as the sequence progresses indefinitely. Understanding this helps in determining whether an infinite series sum results in a finite value.
In our telescoping series example, the sequence of partial sums decreases towards a limit. This is evident as terms like \( -\frac{1}{n+1} \) cancel with subsequent terms, resulting in fewer uncanceled terms as the sequence progresses.

• Recognize convergence by noticing terms disappearing over iterations.
• Observe the behavior of residual terms like \( \lim_{n \to \infty} \frac{1}{n+1} \), which approaches zero.
• Convergence to a specific value, such as 1 in our problem, indicates the sequence's stability and final sum.

Understanding sequence convergence is essential for digging into infinite series, ensuring we correctly find their sum by recognizing limiting behavior. This knowledge is fundamental to evaluating how series sums reach a finite total, as seen in telescoping series.