Summation notation review

Here are the key points for studying "Summation Notation":

1. Definition: Summation notation (Σ) is a concise way to represent the sum of a sequence of numbers.

2. Components:

   • Σ Symbol: Indicates summation.
   • Index of Summation: A variable (usually $i$) that represents each term in the series.
   • Limits of Summation: The lower and upper bounds (e.g., $i = 1$ to $n$) that define the range of the index.
   • Expression: The formula that defines the term being summed (e.g., $i^2$).

3. Basic Structure:

   $$\sum_{i=a}^{b} f(i)$$

   where $a$ is the lower limit, $b$ is the upper limit, and $f(i)$ is the function being summed.

4. Properties of Summation:

   • Linearity: $\sum (c \cdot f(i)) = c \cdot \sum f(i)$, where $c$ is a constant.
   • Sum of Sums: $\sum (f(i) + g(i)) = \sum f(i) + \sum g(i)$.
   • Splitting Summations: $\sum_{i=a}^{b} f(i) = \sum_{i=a}^{c} f(i) + \sum_{i=c+1}^{b} f(i)$ for any $c$ between $a$ and $b$.

5. Common Series:

   • Arithmetic Series: Sum of terms in a linear sequence.
   • Geometric Series: Sum of terms where each term is a fixed multiple of the previous term.

6. Changing the Index: Strategies for manipulating indices, such as substituting $j = i + k$.

7. Applications: Used in various fields like statistics, calculus, and computer science to simplify calculations and express formulas.

By mastering these key points, you can effectively work with summation notation in mathematical contexts.

When studying "Worked Examples: Summation Notation," key points to focus on include:

1. Understanding Summation Notation: Recognize the symbols involved, such as the sigma (Σ), limits of summation (lower and upper bounds), and the index of summation.

2. Setting Up Summations: Learn how to express a series of numbers or a mathematical function in summation form.

3. Applying Properties: Familiarize yourself with properties of summation, such as linearity (breaking a sum into parts) and rules for shifting indices.

4. Evaluating Summations: Practice calculating sums using closed-form formulas for common series (e.g., arithmetic series, geometric series).

5. Transforming Summations: Understand how to manipulate summation expressions, including re-indexing and changing limits.

6. Example Problems: Study various worked examples to see the application of summation notation in solving problems, reinforcing conceptual understanding.

7. Interpreting Results: Learn how to interpret the output of summations in context, ensuring clarity on what the result represents.

By concentrating on these points, you'll build a solid foundation in understanding and using summation notation effectively.