Problem 6 - Calculus

Question

Express these sums in sigma notation: $$ y_{1}-v_{2}+v_{3}-v_{4} \quad v_{1} w_{1}+v_{2} w_{2}+\cdots+v_{n} w_{n} \quad v_{1}+v_{3}+v_{5} $$

Step-by-Step Solution

Answer

\( y_1 + \sum_{i=2}^{3} (-1)^{i} v_{i}; \sum_{i=1}^{n} v_{i} w_{i}; \sum_{i=1}^{3} v_{2i-1} \)

Step 1: Understand the Problem

We are given three different expressions and asked to express them in sigma notation. Sigma notation is a way of writing a sum of terms that are defined by a rule or a pattern.

Step 2: Analyze the First Expression

The first expression is: \( y_{1} - v_{2} + v_{3} - v_{4} \). Notice the alternating signs between the terms and the indices associated with \( v \). This suggests a pattern we can express with a sigma notation.

Step 3: Express the First Sum in Sigma Notation

The expression \( y_{1} - v_{2} + v_{3} - v_{4} \) does not have a consistent pattern for all terms since it starts with \( y_1 \). The rest follows a pattern of adding and subtracting consecutive terms of the sequence \( v \). This can be partially expressed using sigma notation for \( v \):\[ y_1 + \sum_{i=2}^{3} (-1)^{i} v_{i} \]

Step 4: Analyze the Second Expression

The second expression is a standard sum: \( v_{1} w_{1} + v_{2} w_{2} + \cdots + v_{n} w_{n} \). The pattern here is the multiplication of corresponding terms \( v_i \) and \( w_i \).

Step 5: Express the Second Sum in Sigma Notation

The sequence \( v_{1} w_{1} + v_{2} w_{2} + \cdots + v_{n} w_{n} \) can be directly expressed using sigma notation:\[ \sum_{i=1}^{n} v_{i} w_{i} \]

Step 6: Analyze the Third Expression

The third expression is \( v_{1} + v_{3} + v_{5} \). This is an arithmetic sequence involving every other term starting from \( v_1 \).

Step 7: Express the Third Sum in Sigma Notation

The expression \( v_{1} + v_{3} + v_{5} \) can be expressed as the sum of terms with odd indices:\[ \sum_{i=1}^{3} v_{2i-1} \]

Step 8: Review and Finalize the Sigma Notations

Check each expression to ensure they fit a pattern or rule representable in sigma notation. Adjust indices or constants as needed for accuracy.

Key Concepts

Algebraic Expression

An algebraic expression is a combination of variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. It can be as simple as a single term or a complex combination of several terms. Each term in an algebraic expression may contain coefficients, which are the numerical parts that multiply the variables. For instance, in the expression \(3x + 4y\), \(3x\) and \(4y\) are terms, with \(3\) and \(4\) being the coefficients. By understanding these components, students can manipulate and simplify expressions effectively, which is crucial for solving equations and understanding more advanced mathematical concepts.

Summation Notation

Summation notation, also known as sigma notation, is a concise way of writing a sum of a sequence of terms. It uses the Greek letter sigma (\( \Sigma \)) to represent the sum. The general form is:

• \( \sum_{i=a}^{b} f(i) \)

Here, \(f(i)\) is the function representing terms in the sequence, \(i\) is the index of summation which takes integer values from \(a\) to \(b\), inclusive. This notation is handy for expressing repetitive addition in a compact form. For example, the summation \( \sum_{i=1}^{n} i \) represents the sum of all integers from \(1\) to \(n\), equivalent to \(1 + 2 + ... + n\). Understanding summation notation is important for working with arithmetic series and other sequences.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the "common difference." Arithmetic sequences can be represented in both explicit and recursive forms. For example:

• Explicit form: \(a_n = a_1 + (n-1)d\)
• Recursive form: \(a_n = a_{n-1} + d\)

Here, \(a_n\) stands for the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. Understanding arithmetic sequences is essential for recognizing patterns and deriving formulas for sums, such as those representable in sigma notation.

Alternating Series

An alternating series is one in which the terms switch signs alternately. Typically, these series can be expressed as:

• \(a_1 - a_2 + a_3 - a_4 + \ldots\)

The factors that determine the alternation can often be represented using powers of \(-1\). For example, the series can be represented as:

• \( \sum_{i=1}^{n} (-1)^{i+1} a_i \)

Here, \((-1)^{i+1}\) ensures the sign of each term alternates. Understanding alternating series is useful, particularly in calculus and analysis, as they help in solving series convergence problems and working with trigonometric and exponential functions.