Problem 36 - 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. $$-1,1,-1,1, \ldots$$

Step-by-Step Solution

Answer

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

Step 1: Understand the sequence

The given sequence is: \(-1, 1, -1, 1, \ldots\). It continues alternating between \(-1\) and \(1\).

Step 2: Define the first partial sum \(S_1\)

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

Step 3: Calculate the second partial sum \(S_2\)

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

Step 4: Calculate the third partial sum \(S_3\)

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

Step 5: Calculate the fourth partial sum \(S_4\)

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

Step 6: Calculate the fifth partial sum \(S_5\)

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

Step 7: Calculate the sixth partial sum \(S_6\)

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

Key Concepts

Sequence

A sequence is essentially a list of numbers arranged in some particular order. In mathematics, each term in a sequence is associated with a specific position. A sequence can be finite, with a limited number of terms, or infinite, continuing indefinitely. Sequences are often used to represent patterns or relationships in numbers.
In the context of the problem, the sequence is a specific kind called an **alternating sequence**. Here, it continues indefinitely, switching between -1 and 1. The term "sequence" comes into play as we deal with the order in which numbers appear and how they align with each respective position. The position in the sequence helps us find partial sums or other properties associated with the numbers.

Alternating Sequence

An alternating sequence is a type of sequence where the sign of the terms alternates between positive and negative. This means that consecutive terms will differ in sign. In our given example sequence, this is clearly seen as it alternates between -1 and 1:

• The first term is -1.
• The second term is 1.
• The third term is -1 and so on.

This feature of alternating sequences introduces a regular switching pattern, which leads to interesting behaviors when we discuss summation, particularly in partial sums. Understanding this pattern is crucial as it significantly affects how we calculate the sums of these sequences, especially over longer stretches of terms. Alternating sequences can oscillate or converge, depending on their terms and characteristics.

Summation

Summation is the process of adding terms together. In mathematics, summation provides a way to combine sequence terms into a single value. With our alternating sequence, the process of summation gives rise to what are called partial sums.
Partial sums are intermediate sums taken from a sequence. For example, the first partial sum is simply the first term. Subsequent partial sums include additional terms from the sequence. Applying summation to an alternating sequence results in patterns:

• The first partial sum, \(S_1\), is \(-1\).
• The second partial sum, \(S_2\), becomes 0, since \(-1 + 1 = 0\).
• This pattern continues with \(S_3 = -1\), \(S_4 = 0\), and so on.

This pattern is due to each pair of consecutive terms, with opposing signs, canceling each other out. Understanding this alternation of sign is key to comprehending the partial sums of alternating sequences.