Problem 46 - Calculus: Early Transcendentals

Question

Find all values of \( c \) for which the following series converges. \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac {c}{n} - \frac {1}{n + 1} \right) \)

Step-by-Step Solution

Answer

The series converges for \( c = 1 \).

Step 1: Understanding the Series

We have the series \( \sum_{n=1}^{\infty} \left( \frac{c}{n} - \frac{1}{n+1} \right) \). We need to analyze the convergence of this series for all possible values of \( c \).

Step 2: Rewrite the Terms

Let's rewrite the general term as \( a_n = \frac{c}{n} - \frac{1}{n+1} \). We can consider dividing these fractions into simpler forms when finding convergence.

Step 3: Simplify the Expression

The expression \( \frac{c}{n} - \frac{1}{n+1} \) can be combined using a common denominator: \( \frac{c(n+1) - n}{n(n+1)} = \frac{cn + c - n}{n(n+1)} \).

Step 4: Analyze Convergence of Each Part

Consider the series split into two parts: \( \sum \frac{c}{n} \) and \( -\sum \frac{1}{n+1} \). The series \( \sum \frac{1}{n} \) and its shifted version \( \sum \frac{1}{n+1} \) are well-known as the harmonic series, which diverges.

Step 5: Evaluate Convergence Conditions

For convergence, the \( \frac{c}{n} \) part must present a scenario such that the entire series converges. Knowing both \( \sum \frac{1}{n} \) and \( \sum \frac{1}{n+1} \) diverge similarly, their combination forms a \( \ln(n) \) behavior. We require \( c = 1 \) to ensure the cancellation leads to convergence.

Step 6: Conclusion

The entire series converges if and only if the constant \( c \) results in the predominant terms cancelling each other out. This occurs when \( c = 1 \), making the form \( \sum \left( \frac{1}{n} - \frac{1}{n+1} \right) \), known to be telescoping.

Key Concepts

Harmonic Series

The harmonic series is a fascinating and fundamental concept in calculus. It is given by the sum \( \sum_{n=1}^{\infty} \frac{1}{n} \). To better understand its behavior, imagine adding fractions like \( \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots \). Even though each individual fraction becomes smaller as \( n \) increases, the sum does not stabilize but rather grows indefinitely.

This characteristic means that the harmonic series diverges. Simply put, it doesn't settle at a finite value as you add more terms. A helpful way to visualize this is to consider pulling pebbles from an endless heap—no matter how small each handful gets, the heap never runs out.

• The key point: Divergence means no matter how many terms you add, you'll never reach a final sum.
• The essence: The terms shrink, but the series never stops growing.

Understanding the harmonic series is crucial when analyzing series like the one in the exercise, where certain parts mimic this divergent behavior.

Telescoping Series

A telescoping series is a special type of series where many terms cancel each other out in succession. For instance, imagine a collapsing telescope—you can think of the series terms canceling just like those sliding sections. The result is a series that, instead of going off to infinity like the harmonic series, has a finite sum.

For example, take \( \sum \left( \frac{1}{n} - \frac{1}{n+1} \right) \). Here, most intermediate terms vanish when expanded. This significant reduction leads to easier convergence conditions.

• Key Feature: The terms vanish or cancel out pairwise, simplifying the sum considerably.
• Convergence: Often guarantees a finite limit, contrasting with the divergent nature of the harmonic series.

Applying this to the exercise, recognizing the potential cancellation in the expression provided suggests a condition where it can become a telescoping series—specifically, when \( c = 1 \). That's when the series becomes perfectly telescoping, thus converging.

Infinite Series

An infinite series is simply a series that extends indefinitely, adding up an infinite number of terms. Whether these terms combine to form a finite, divergent, or oscillating sum is of great interest in calculus.

The convergence or divergence of an infinite series depends on the behavior of its terms as the series progresses. Analyzing convergence typically involves looking for patterns or behaviors that either stabilize (converge) or grow without bound (diverge).

• Convergent Series: Their sum approaches a specific value as more terms are added.
• Divergent Series: Their sum does not settle, like the harmonic series we discussed earlier.
• Tools: Tests like the ratio test, comparison test, and recognizing forms such as telescoping help in determining the nature of a series.

In the solution provided, recognizing separation into known divergent parts (harmonic series) and finding conditions to form a convergent series (telescoping), directly assesses the nature of infinite series.