Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 13-22, use any test developed so far, including any from Section 9.2, to decide about the convergence or divergence of the series. Give a reason for your conclusion. \(\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{k+1}\right)\)

Step-by-Step Solution

Answer

The series converges to 1 by telescoping.

Step 1: Identify the Series Type

The given series is \( \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{k+1} \right) \) which is a telescoping series. In telescoping series, consecutive terms cancel each other such that only a few terms are left uncanceled.

Step 2: Partial Sum Simplification

Consider the partial sum \( S_n = \sum_{k=1}^{n} \left( \frac{1}{k} - \frac{1}{k+1} \right) \). Writing out the first few terms: \( \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \cdots + \left( \frac{1}{n} - \frac{1}{n+1} \right) \). Notice that the terms \( -\frac{1}{2}, -\frac{1}{3}, \ldots, -\frac{1}{n} \) cancel with positive terms from subsequent fractions.

Step 3: Evaluate the Remaining Terms

After cancellation, the only terms that do not have a counterpart are \( \frac{1}{1} \) and \( -\frac{1}{n+1} \). Hence the partial sum becomes \( S_n = 1 - \frac{1}{n+1} \).

Step 4: Determine the Limit of the Partial Sum

Calculate the limit of the sequence of partial sums as \( n \to \infty \): \[ \lim_{n \to \infty} S_n = \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right) = 1. \] Since the limit of the partial sums is finite, the series converges.

Step 5: Conclusion

The series \( \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{k+1} \right) \) converges because the sequence of partial sums tends to a finite limit, which is 1.

Key Concepts

Telescoping Series

A telescoping series is a special kind of series where most terms cancel out between consecutive terms, allowing for simplification. The key characteristic of a telescoping series is that after expansion and simplification, the majority of the middle terms cancel out, leaving just a few terms to evaluate.
In the given series, \( \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{k+1} \right) \), if we write out the terms, we see a pattern: - \( \frac{1}{1} - \frac{1}{2} \)- \( \frac{1}{2} - \frac{1}{3} \)- \( \ldots \)- \( \frac{1}{n} - \frac{1}{n+1} \)Here, the middle terms cancel, such as \(-\frac{1}{2}, +\frac{1}{2}\), resulting in a much simpler expression involving just a few terms. This property makes telescoping series easier to sum when deciding convergence.

Partial Sums

The concept of partial sums is vital in understanding how a series behaves as it approaches infinity. A partial sum \( S_n \) of a series is the sum of its first \( n \) terms. This gives us a way to approximate and examine the entire series by looking at smaller, finite parts.
For the telescoping series \( \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{k+1} \right) \), we form the partial sum:\[ S_n = \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \cdots + \left( \frac{1}{n} - \frac{1}{n+1} \right) = 1 - \frac{1}{n+1} \]Here, after cancelation we only have the first and the last terms remaining, illustrating the telescoping property. This simplification is crucial for understanding the convergence of the series.

Limit of Sequence

The limit of a sequence is a fundamental concept in understanding series convergence. It tells us what value the sequence of numbers approaches as the number of terms grows indefinitely. A series is said to converge if the sequence of its partial sums approaches a finite limit.
For our given series, the partial sum is \( S_n = 1 - \frac{1}{n+1} \). To determine convergence, we compute:\[ \lim_{n \to \infty} S_n = \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right) = 1 \]This shows that as \( n \) increases, the partial sums approach 1, indicating the series converges to this value. This convergence to a finite number, 1, is the defining characteristic of a convergent series.

Convergence Tests

Convergence tests are methods used to determine whether a series converges or diverges. For different types of series, there are numerous tests available, including comparison test, ratio test, integral test, and specific tests for series like telescoping and alternating series.
In the context of telescoping series like \( \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{k+1} \right) \), the telescoping property itself acts as a test for convergence. By simplifying the series into partial sums and investigating the limit as \( n \to \infty \), we observe that the series converges to a finite limit, verifying its convergence.
These tests are powerful tools that help us deal with different types of series and ensure we can accurately assess their behavior as they extend to infinity.