Question

Are the expressions in Problems equivalent? \(\sum_{i=1}^{20}(-1)^{i} b_{n}\) and \(-b_{1}+b_{2}+b_{3}+\cdots+b_{19}+b_{20}\)

Short answer

Answer: Yes, the two expressions are equivalent.

Step-by-step solution

Understand the first expression

The first expression \(\sum_{i=1}^{20}(-1)^{i} b_{n}\) represents a sum of 20 terms, where each term is of the form \((-1)^{i} b_{n}\). Notice that the exponent of \((-1)\) is equal to the index \(i\) in the sum, which means that we alternate between positive and negative terms.

Expand the first expression

Let's expand the first expression to understand its terms better. \(\sum_{i=1}^{20}(-1)^{i} b_{n} = (-1)^1 b_1 + (-1)^2 b_2 + (-1)^3 b_3 + \cdots + (-1)^{19} b_{19} + (-1)^{20} b_{20}\).

Simplify the first expression

We can now simplify this expression by recognizing that \((-1)^{odd}\) will always be \(-1\) and \((-1)^{even}\) will always be \(1\). The expression now becomes: \(-b_1 + b_2 - b_3 + \cdots - b_{19} + b_{20}\).

Compare the second expression

The second expression is given as \(-b_{1}+b_{2}+b_{3}+\cdots+b_{19}+b_{20}\). We can notice that the terms in both the expressions match. Based on the above steps, we can conclude that the two given expressions, \(\sum_{i=1}^{20}(-1)^{i} b_{n}\) and \(-b_{1}+b_{2}+b_{3}+\cdots+b_{19}+b_{20}\), are equivalent.

Key concepts

Sigma Notation

Sigma notation, denoted as \( \sum \), is a concise way of writing sums, particularly those with many terms. It represents the summation of a sequence of numbers or expressions. When you see sigma notation, the expression underneath, above, and beside the sigma will guide you on what to sum.

• Below the sigma is the starting index, in this case, \( i=1 \).
• Above the sigma is the stopping index, which is \( 20 \) in the exercise.
• Beside the sigma is the expression to be summed, here, \((-1)^{i} b_{n}\).

This notation is especially useful for compactly representing long summations, preventing the need to write out each individual term.
In our problem, sigma notation helps easily express the process of alternating the terms as we switch each factor from negative to positive based on the exponent \( i \), which simplifies complex expressions and aids comprehension.

Alternating Series

An alternating series is a sequence of numbers where the terms are alternately positive and negative. It's easy to identify because the signs "alternate," or switch back and forth.

• For the given exercise, \((-1)^{i}\) determines the sign of each term. If \( i \) is odd, \((-1)^{i} = -1\), making the term negative.
• If \( i \) is even, \((-1)^{i} = 1\), making the term positive.

This creates an alternating series of terms like \(-b_1 + b_2 - b_3 + \cdots\).
Alternating series are common in mathematical analysis and play a significant role in calculus, especially in series convergence and approximation of functions. Being able to identify the alternating pattern is key to understanding and simplifying expressions.

Simplification of Expressions

Simplifying expressions involves rewriting them in a way that is easily understandable and more concise without changing their value.
In mathematical problems, especially with sigma notation exercises, simplification can reveal equivalent expressions that look different at first glance.

• The original expression \( \sum_{i=1}^{20} (-1)^{i} b_{n} \) simplifies by recognizing that \((-1)\) raised to odd and even powers will alternate the terms' signs.
• Through simplification, it transforms into \(-b_1 + b_2 - b_3 + \ldots - b_{19} + b_{20}\).

Recognizing patterns in such problems allows for clear comparisons between supposedly different expressions, guiding to the conclusion that, indeed, they are equivalent.