Problem 49 - Thomas Calculus

Question

Find the sum of each series in Exercises \(45-52 .\) $$\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$$

Step-by-Step Solution

Answer

The sum of the series is 1.

Step 1: Analyze the Telescoping Series

The given series is \( \sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right) \).This is an example of a telescoping series where each term can be expanded \( a_n = \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} \) so that it simplifies when summed.

Step 2: Write the First Few Partial Sums

Write out the first few terms of the sum:\[ S_1 = \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{2}} \]\[ S_2 = \left(\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{2}}\right) + \left(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{3}}\right) \]\[ S_3 = \left(\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{2}}\right) + \left(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{3}}\right) + \left(\frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}}\right) \]Each time, intermediate terms cancel out, leaving \( S_n = 1 - \frac{1}{\sqrt{n+1}} \).

Step 3: Compute the Limit of the Partial Sum

The finite sum of the first \( n \) terms, \( S_n = 1 - \frac{1}{\sqrt{n+1}} \), needs to be evaluated as \( n \to \infty \).As \( n \to \infty \), \( \frac{1}{\sqrt{n+1}} \to 0 \).Therefore, the limit is:\[ \lim_{n \to \infty} S_n = 1 - 0 = 1 \]

Key Concepts

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. Imagine a never-ending list of numbers being added together. In mathematical terms, an infinite series is typically denoted as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the general term of the series. When considering whether an infinite series has a definite sum, not all do. Some series might grow indefinitely or oscillate without settling on a single value. Solving these series requires careful mathematical techniques.
One particular type of infinite series is the telescoping series, where successive terms cancel each other, simplifying the series. This was explored in the original exercise, which highlighted how the telescoping feature makes it possible to find the sum.

Partial Sums

The concept of partial sums involves summing up a finite number of terms in a series. For an infinite series, the partial sum, \( S_n \), is the sum of its first \( n \) terms: \( S_n = a_1 + a_2 + \ldots + a_n \). Studying partial sums helps understand the behavior of the series before attempting to sum all its terms.
In the telescoping series example, partial sums revealed the simplification that occurs, as intermediate terms canceled each other out. By simplifying the terms this way, one can quickly see the pattern and deduce a simple expression for \( S_n \).

• Example: The series term \( a_n = \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} \) leads to cancellation when summed.
• Result: \( S_n = 1 - \frac{1}{\sqrt{n+1}} \).

Partial sums also provide a stepping stone towards examining convergence.

Convergence of Series

For an infinite series to be useful in practical applications, it must converge. This means that as more terms are added, the partial sums approach a specific value. Convergence determines whether an infinite sum is meaningful.
In our telescoping series example, the limit of the partial sum \( S_n = 1 - \frac{1}{\sqrt{n+1}} \) was computed as \( n \rightarrow \infty \). Here, \( \frac{1}{\sqrt{n+1}} \rightarrow 0 \) as \( n \to \infty \), which ensures that the series converges to 1.

• Convergence is often shown by taking the limit of the partial sum.
• If the limit of \( S_n \) exists and is finite, the series converges.

Understanding convergence is crucial in ensuring an infinite series provides a finite and reliable sum.

Mathematical Induction

Mathematical induction is a powerful method of proof used to establish the truth of an infinite sequence of statements. It works by proving a base case and showing that if one statement is true, then the next one in the sequence holds true as well. Though not directly used in the exercise, induction is closely related to understanding series.
For example, once a pattern is observed in a telescoping series, induction might be used to confirm that this pattern holds for all terms.

• Base Case: Verify the statement for the initial term.
• Inductive Step: Assume the statement holds for \( n = k \), and then show it holds for \( n = k+1 \).

While the series exercise used a direct computation approach, induction often underlies the logical structure in validating mathematical constructs over a set of numbers.