Problem 37 - Precalculus: Mathematical for Calculus

Question

Find the first four partial sums and the \(n\) th partial sum of the sequence \(a_{n}\) $$a_{n}=\sqrt{n}-\sqrt{n+1}$$

Step-by-Step Solution

Answer

The first four partial sums are \(1-\sqrt{2}\), \(1-\sqrt{3}\), \(-1\), and \(1-\sqrt{5}\). The \(n\)th partial sum is \(1-\sqrt{n+1}\).

Step 1: Calculate First Partial Sum

The first partial sum, denoted as \(S_1\), is simply the first term of the sequence. Thus,\[ S_1 = a_1 = \sqrt{1} - \sqrt{2} = 1 - \sqrt{2}. \]

Step 2: Calculate Second Partial Sum

The second partial sum, \(S_2\), is the sum of the first two terms of the sequence.\[ S_2 = a_1 + a_2 = (\sqrt{1} - \sqrt{2}) + (\sqrt{2} - \sqrt{3}). \] Simplifying this expression,\[ S_2 = \sqrt{1} - \sqrt{3} = 1 - \sqrt{3}. \]

Step 3: Calculate Third Partial Sum

The third partial sum, \(S_3\), includes the sum of the first three terms:\[ S_3 = a_1 + a_2 + a_3 = (\sqrt{1} - \sqrt{2}) + (\sqrt{2} - \sqrt{3}) + (\sqrt{3} - \sqrt{4}). \] This simplifies to:\[ S_3 = \sqrt{1} - \sqrt{4} = 1 - 2 = -1. \]

Step 4: Calculate Fourth Partial Sum

The fourth partial sum, \(S_4\), is the sum of the first four terms of the sequence:\[ S_4 = a_1 + a_2 + a_3 + a_4 = (\sqrt{1} - \sqrt{2}) + (\sqrt{2} - \sqrt{3}) + (\sqrt{3} - \sqrt{4}) + (\sqrt{4} - \sqrt{5}). \] This simplifies to:\[ S_4 = \sqrt{1} - \sqrt{5} = 1 - \sqrt{5}. \]

Step 5: Derive General Formula for the n-th Partial Sum

To find the \(n\)th partial sum, \(S_n\), we sum the first \(n\) terms:\[ S_n = a_1 + a_2 + \ldots + a_n. \] Observing that terms cancel out in pairs (telescoping series), the sequence simplifies to:\[ S_n = \sqrt{1} - \sqrt{n+1}. \] The last term \(\sqrt{n+1}\) does not get canceled.

Key Concepts

Telescoping Series

A telescoping series is a special type of series where many terms cancel out with each other, leaving only a few terms that affect the result. This makes them particularly interesting and often easier to work with. In the given example, the series is defined by the terms \( a_n = \sqrt{n} - \sqrt{n+1} \).

Let's see how a telescoping series works:

• When you add consecutive terms: \( (\sqrt{1} - \sqrt{2}) + (\sqrt{2} - \sqrt{3}) + \cdots +(\sqrt{n} - \sqrt{n+1}) \), the inner terms cancel each other out.
• The terms \( \sqrt{2}, \sqrt{3}, \cdots \sqrt{n} \) all cancel with the similar terms from the next or previous sequence.
• You are left with only limited terms from the very start and end: \( \sqrt{1} - \sqrt{n+1} \). This makes it easy to compute larger sums without adding numerous terms separately.

This simplifying nature of telescoping series is what makes problems involving them more approachable and often more elegant to solve.

Sequence and Series

In mathematics, understanding sequences and series is crucial as they are the foundation for calculus and many other areas. A **sequence** is an ordered list of numbers, and a **series** is the sum of a sequence of terms. Let's unravel these concepts a bit more:

• A sequence \( a_1, a_2, a_3, \ldots, a_n \) is simply a list of numbers with a specific pattern or rule for determining each term.
• A series, on the other hand, is the summation of the sequence: \( S_n = a_1 + a_2 + a_3 + \ldots + a_n \).

Partial sums are used to find the sum of the first few terms of a series. This can help determine whether the series converges to a limit or not.
Each term in our sequence is \( a_n = \sqrt{n} - \sqrt{n+1} \), which illustrates how individual terms contribute to the overall sum of a telescoping series. Evaluating these sums can help us understand the behavior of the series as it extends to infinity.

General Formula for nth Term

Finding a general formula for the nth term is crucial in analyzing the behavior of both sequences and series. It enables you to predict any term in the sequence without calculating all preceding terms.

For the exercise provided, we see that:

• The nth term of the sequence is \( a_n = \sqrt{n} - \sqrt{n+1} \).
• The general formula for the nth partial sum \( S_n \) can be concluded as \( S_n = \sqrt{1} - \sqrt{n+1} \).

This simplification shows the power of general formulas in reducing complex problems into simple expressions. By understanding such formulas, students can easily determine the nth partial sum without adding each term separately.
Furthermore, with telescoping series, the general formula reveals that many intermediate terms get cancelled, and only a few influence the final answer. Mastering this concept is key in problems where quick computation of large series sums is required.