Problem 55 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right) $$

Step-by-Step Solution

Answer

The given series converges and the sum of the series is \(3/2\).

Step 1: Split the Series

The first step is to split the given series into two simpler series. Given series is: \[\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right)\] This can be written as: \[\sum_{n=1}^{\infty}\frac{1}{n} - \sum_{n=1}^{\infty}\frac{1}{n+2}\]

Step 2: Shift the Index of the Second Series

The second step is to shift the index of the second series as it starts from \(n+2\). So, let \(n+2 = m\). When \(n = 1\), \(m\) starts from \(3\). Then, \(\sum_{n=1}^{\infty}\frac{1}{n+2}\) becomes \(\sum_{m=3}^{\infty}\frac{1}{m}\)

Step 3: Find the Difference of the Two Series

Subtract the second series from the first one, \[\sum_{n=1}^{\infty}\frac{1}{n} - \sum_{m=3}^{\infty}\frac{1}{m} = \frac{1}{1} + \frac{1}{2} - \sum_{m=2}^{\infty}\frac{1}{m}\]This simplifies to \(1 + 1/2 - 0 = 3/2\)

Step 4: Conclude the Convergence/Divergence

Since the infinite series simplifies to a finite constant, we can conclude that the series converges and its limit is that finite constant.

Key Concepts

Infinite Series

An infinite series is a sum of infinitely many terms. In mathematical notation, it is represented as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) denotes the terms of the series. The concept of an infinite series extends the idea of adding numbers beyond finite boundaries.

• Each term \( a_n \) represents a particular value in the sequence that forms the series.
• The series can either converge to a specific value or diverge without bound.

An important aspect of infinite series is determining whether they converge or diverge. Converging series approach a fixed value as more terms are added, whereas diverging ones do not settle into a single value.Understanding infinite series is crucial in calculus and mathematical analysis. They bridge the gap between finite and infinite reasoning, allowing mathematicians to work with sums that would otherwise be impossible to compute.

Telescoping Series

A telescoping series is a special form of an infinite series. Each term in the series cancels out a part of another term, leading to a clear and often simple sum.

• The name 'telescoping' comes because, like a telescope, the series collapses into a compact form.
• This cancellation property makes telescoping series particularly easy to evaluate.

In our example \( \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right) \), after proper manipulation and shifting of indices, most terms cancel out. Consequently, the series simplifies greatly, leading us to an expression in terms of simple fractions.Using telescoping techniques, we found that the series converges to \( \frac{3}{2} \). This is because the infinite nature of terms diminishes upon simplification, and only a finite number of terms remain significant.

Convergence Tests

Convergence tests are methods used to determine whether an infinite series converges or diverges. These methods equip us with tools to handle the infinite nature of series in a precise way.

• Comparison Test: Compares the series with a known benchmark series.
• Ratio Test: Examines the limit of the ratio of consecutive terms.
• Integral Test: Uses integrals to analyze the behavior of series.

These methods allow mathematicians to judge the behavior of complex series. We can apply these methods systematically to handle the intricacies of various forms of infinite series. In some cases, like with telescoping series, visual inspection and manipulation might be more revealing. However, standard convergence tests provide a generalized framework for approaching series whose convergence is not immediately obvious.