Problem 42 - Precalculus: Mathematics for Calculus

Question

Find the first six partial sums \(S_{1}, S_{2}, S_{3}\), \(S_{4}, S_{5}, S_{6}\) of the sequence whose \(n\)th term is given. \(-1,1,-1,1, \ldots\)

Step-by-Step Solution

Answer

The first six partial sums are: -1, 0, -1, 0, -1, 0.

Step 1: Identify the Sequence

The sequence given is \(-1, 1, -1, 1, \ldots\). This is a repeating sequence where each odd term is \(-1\) and even term is \(1\).

Step 2: Calculate First Partial Sum (S1)

The first partial sum, \(S_1\), is simply the first term of the sequence. So, \(S_1 = -1\).

Step 3: Calculate Second Partial Sum (S2)

The second partial sum, \(S_2\), is the sum of the first two terms: \(-1 + 1 = 0\). Therefore, \(S_2 = 0\).

Step 4: Calculate Third Partial Sum (S3)

The third partial sum, \(S_3\), is the sum of the first three terms: \(-1 + 1 - 1 = -1\). Thus, \(S_3 = -1\).

Step 5: Calculate Fourth Partial Sum (S4)

The fourth partial sum, \(S_4\), is the sum of the first four terms: \(-1 + 1 - 1 + 1 = 0\). Therefore, \(S_4 = 0\).

Step 6: Calculate Fifth Partial Sum (S5)

The fifth partial sum, \(S_5\), is the sum of the first five terms: \(-1 + 1 - 1 + 1 - 1 = -1\). So, \(S_5 = -1\).

Step 7: Calculate Sixth Partial Sum (S6)

The sixth partial sum, \(S_6\), is the sum of the first six terms: \(-1 + 1 - 1 + 1 - 1 + 1 = 0\). Thus, \(S_6 = 0\).

Key Concepts

Sequence

A sequence is a set of numbers arranged in a specific order. In mathematics, sequences may display a repetitive or predictable pattern. Notably, each number in a sequence is called a term, denoted often with subscripts, such as the first term being \(a_1\), the second \(a_2\), and so on. In the example given, the sequence is

• \(-1, 1, -1, 1, \ldots\)

Here, the sequence alternates between \(-1\) and \(1\). This means that it continues infinitely, following the same two-term pattern without change. Understanding sequences is crucial because they are the foundation for more complex mathematical concepts like series and functions.

Alternating Series

An alternating series is a type of series where the terms alternate in sign. Each term is either positive or negative, and typically each term changes sign from the previous one. Alternating series are essential because they behave uniquely, especially regarding convergence, a topic we will discuss later.

• In our given example, the sequence
  • \(-1, 1, -1, 1, \ldots\)
  can naturally be expanded into an alternating series.

This type of series occurs frequently in mathematical situations where the pattern causes terms to repeatedly subtract and add to each other. This behavior makes alternating series useful not only in pure mathematics but also in practical applications like signal processing or mechanical engineering.

Series Convergence

The concept of series convergence refers to whether a series approaches a specific value as more terms are added. For a series to converge, the sequence of its partial sums must approach a finite limit.
For example, in some series, the partial sums get closer and closer to a single number, no matter how many terms are added. However, our example,

• \(-1, 1, -1, 1, \ldots\)

does not converge to a limit. As seen through the partial sums,

• \(S_1 = -1\), \(S_2 = 0\), \(S_3 = -1\), ...,

the partial sums oscillate between \(-1\) and \(0\). This means the series diverges, indicating it does not settle towards any particular value.

Summation

Summation is the process of adding a sequence of numbers or terms in a series. In mathematics, summation helps to find the total of elements in a sequence, calculated through what is known as partial sums.

• Partial sums involve taking subsets of terms from a sequence and calculating their total. For instance, the first few partial sums of our sequence
  • \(-1, 1, -1, 1, -1, 1, \ldots\)
  are computed as follows:
  • \(S_1 = -1\), \(S_2 = 0\), \(S_3 = -1\), \(S_4 = 0\), \(S_5 = -1\), \(S_6 = 0\),

Summation is particularly vital in calculus and analysis, where understanding the behavior of sums of sequences leads to insights on convergence and divergence, integral calculus, and other advanced topics.