Problem 36 - Algebra and Trigonometry

Question

\(33-36\) . Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$ -1,1,-1,1, \dots $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Sequence

The sequence is given as \(-1, 1, -1, 1, ext{...}\). It is an alternating sequence where each term switches between -1 and 1.

Step 2: Understand Partial Sums

The partial sum \(S_n\) is defined as the sum of the first \(n\) terms of the sequence. Therefore, \(S_1\) is the sum of the first term, \(S_2\) is the sum of the first two terms, and so on.

Step 3: Calculate the First Partial Sum \(S_1\)

\(S_1 = -1\). This is just the first term of the sequence.

Step 4: Calculate the Second Partial Sum \(S_2\)

We add the first two terms:\[ S_2 = -1 + 1 = 0 \]

Step 5: Calculate the Third Partial Sum \(S_3\)

We sum the first three terms:\[ S_3 = -1 + 1 - 1 = -1 \]

Step 6: Calculate the Fourth Partial Sum \(S_4\)

We sum the first four terms:\[ S_4 = -1 + 1 - 1 + 1 = 0 \]

Step 7: Calculate the Fifth Partial Sum \(S_5\)

We sum the first five terms:\[ S_5 = -1 + 1 - 1 + 1 - 1 = -1 \]

Step 8: Calculate the Sixth Partial Sum \(S_6\)

We sum the first six terms:\[ S_6 = -1 + 1 - 1 + 1 - 1 + 1 = 0 \]

Key Concepts

Alternating Sequence

When we talk about alternating sequences, we're referring to a type of sequence where the signs of the terms switch back and forth. In the given sequence

• -1, 1, -1, 1,...

this swaying pattern is evident. Here, every odd-positioned term is > -1 and every even-positioned term is > 1. Alternating sequences often arise in mathematical series and even in real-world phenomena.
To construct an alternating sequence, you usually have a pattern in the exponents or > multiplier terms. For instance, a term like > (-1)^n gives us alternating values: when > n is odd, > (-1)^n = -1 ; when > n is even, > (-1)^n = 1. Making these sequences predictable and fun to work with. Understanding this alternating nature is key when calculating partial sums or dealing with series involving these sequences.

Partial Sum Formulas

A partial sum is the sum of a specified number of initial terms in a sequence. For an alternating sequence, like our example, identifying partial sums involves adding together only as many terms as needed for that specific partial sum.For instance,

• Partial sum > \( S_2 \) covers the first two terms (-1 and 1).
• And, Partial sum > \( S_4 \) includes the first four terms (-1, 1, -1, 1).

To explore the formula involved, consider the understanding of > sum up to even and odd terms. For alternating sequences:

• If > \( n \) is odd, > \( S_n \) tends to >−1.
• > If \( n \) is even, > \( S_n \) results in >0.

This comes in handy as a regular pattern, helping to predict whether the partial sum will be zero or non-zero. Breakdown of calculations provides intuitive output for pattern recognition in alternating sequences.

Summation of Series

In mathematics, a series is the sum of the terms of a sequence. When we sum up all terms indefinitely, it is ideally termed an infinite series. However, summation in this topic of partial sums deals only with a finite number of terms. Observe how we managed to break down summation decisions using partial sums. It's crucial to remember that with alternating sequences, variability lies in values constantly reversing. Summation helps you comprehend:

• The behavior of series, especially when it converges towards a fixed number.
• Building connections between infinite series and their finite counterparts (i.e., partial sums).

When extending knowledge from the sequence to the series, remember that techniques such as telescoping series or geometric series calculations could become relevant, though they're beyond basic purposes of partial summation actions. By practicing, gradually, these alternating partial sums offer good insights into convergence, divergence, and summation techniques.