7: Sequences and Series, Mathematical Induction, and the Binomial Theorem
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- 7.1: Sequences
- This section introduces sequences, defining them as ordered lists of numbers generated by a function. It covers different types of sequences, including arithmetic and geometric sequences, and explains how to write them explicitly and recursively. The section also explores convergence and divergence of sequences, providing examples and methods to determine their behavior. It serves as a foundation for understanding series and other advanced topics.
- 7.2: Summation Notation
- This section introduces summation notation, also known as sigma notation, which is used to represent the sum of terms in a sequence. It explains the structure of summation notation, including the lower and upper limits of summation and the index of summation. Examples demonstrate how to interpret and calculate sums using this notation, making it a powerful tool for expressing long sums in a concise form.
- 7.3: Mathematical Induction
- This section introduces mathematical induction, a proof technique used to establish statements for all natural numbers. It explains the process in two steps: proving a base case and showing that if the statement holds for one integer, it also holds for the next. Examples illustrate how to apply induction to verify formulas and inequalities, providing a foundation for proving mathematical statements systematically.
- 7.4: The Binomial Theorem
- This section introduces the Binomial Theorem, which provides a formula for expanding binomials raised to a power. It explains how to use binomial coefficients and Pascal's Triangle to expand expressions like (a+b)n. The section also covers combinations and includes examples demonstrating how to apply the theorem for different powers, making it useful for simplifying polynomial expansions.
Thumbnail: The sum of the areas of the rectangles is greater than the area between the curve f(x)=1/x and the x-axis for x≥1. Since the area bounded by the curve is infinite (as calculated by an improper integral), the sum of the areas of the rectangles is also infinite.