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Monotonic Increasing Sequences Natural Numbers Importance of natural numbers in the context of monotonic increasing sequences: Natural numbers are often used as the indexing set for sequences. Monotonic increasing sequences are typically defined in terms...

Monotonic Increasing Sequences Natural Numbers Importance of natural numbers in the context of monotonic increasing sequences: Natural numbers are often used as the indexing set for sequences. Monotonic increasing sequences are typically defined in terms of natural numbers. Geometric Progression A sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number. Example: 1, 3, 9, 27,... (geometric progression with common ratio 3) Monotonic Increasing Sequences Definition: A sequence {a_n} is monotonic increasing if a_n ≤ a_{n+1} for all n in the natural numbers. Examples: {1, 2, 3,...} {2, 4, 6,...} Convergence of Sequences A sequence {a_n} converges to a limit L if for any ε > 0, there exists a natural number N such that for all n ≥ N, |a_n - L| < ε. Monotonic increasing sequences that are bounded above converge to their supremum. Bounded Sequences A sequence {a_n} is bounded if there exists a number M such that |a_n| ≤ M for all n in the natural numbers. An unbounded sequence diverges to infinity or negative infinity. Limit of a Sequence The limit of a sequence is the value that the sequence approaches as the index n approaches infinity. Not all sequences have a limit. Limitations of Infinity in Math Infinity is not a number, but a concept used to describe something that is unbounded or without end. Infinity cannot be used as a limit in mathematical proofs without careful consideration. For example, the sequence {1, 2, 3,...} does not have a limit, even though it goes to infinity. Calculating Limits to Infinity  To calculate the limit of a sequence as n goes to infinity: 1. Find the terms of the sequence. 2. Look for a pattern in the terms. 3. Use algebraic manipulation to find the limit. Example: Calculate the limit of the sequence {1/n} as n goes to infinity. 1. Terms: 1, 1/2, 1/3, 1/4,... 2. Pattern: Each term is 1 divided by a larger and larger number. 3. Limit: 0 Understanding Sequence Expansion Sequence expansion is a way to generate new sequences from existing ones. Example: Given the sequence {1, 2, 3,...}, expand it to {1, 2, 3, 4, 5,...} by adding a new term at the end. Concept of Monotonic Increasing A sequence is monotonic increasing if each term is greater than or equal to the previous term. The concept of monotonic increasing is used in the definition of monotonic increasing sequences. Finding Limits of a Function The limit of a function f(x) as x approaches a is the value that the function approaches as x gets arbitrarily close to a. The limit of a function is different from the value of the function at a point. Sequence Convergence Theorems Theorems about the convergence of sequences, such as: If a sequence converges, then it is bounded. If a monotonic increasing sequence is bounded above, then it converges to its supremum. Converting to Denominator Form Converting a fraction to denominator form involves writing the fraction as a/b, where a and b are integers and b is positive. Not directly related to monotonic increasing sequences, but a common mathematical concept. Monotonic Decreasing Sequences Definition: A sequence {a_n} is monotonic decreasing if a_n ≥ a_{n+1} for all n in the natural numbers.  Not directly related to monotonic increasing sequences, but a similar concept. Convergence of Sequences Notes Importance of Natural Numbers Natural numbers are the foundation of most mathematical concepts and theories. They are used to define sequences and their properties. Geometric Progression A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number. This number is known as the common ratio. Converting to Denominator Form Converting sequences to denominator form can help in understanding and analyzing their convergence properties. Monotonic Decreasing Sequences A sequence is said to be monotonic decreasing if each term is smaller than the previous term. The convergence of a monotonic decreasing sequence is closely related to its infimum (greatest lower bound). Sequence Convergence Theorems There are several theorems that provide necessary and sufficient conditions for the convergence of sequences, such as the Monotone Convergence Theorem and the Bolzano- Weierstrass Theorem. Understanding Sequence Expansion Understanding how sequences expand and grow can aid in determining their convergence or divergence properties. Monotonic Increasing Sequences A sequence is said to be monotonic increasing if each term is greater than the previous term. The convergence of a monotonic increasing sequence is closely related to its supremum (least upper bound). Calculating Limits to Infinity The limit of a sequence as it approaches infinity is a fundamental concept in the convergence of sequences. Convergence of Sequences A sequence is said to converge if the values of the sequence terms approach a finite limit as the sequence progresses. Bounded Sequences A sequence is said to be bounded if there exist numbers M and N such that every term of the sequence is between M and N. Limit of a Sequence The limit of a sequence is the value that the sequence terms approach as the sequence progresses. Finding Limits of a Function Finding the limit of a function as it approaches infinity is similar to finding the limit of a sequence. Concept of Monotonic Increasing The concept of monotonic increasing sequences plays a crucial role in understanding and analyzing the convergence of sequences. Limitations of Infinity in Math It is important to understand the limitations of infinity in mathematics, as it can lead to erroneous conclusions. Note: This notes aims to cover the key concepts related to convergent sequences, with a focus on understanding the necessary and sufficient conditions for convergence and the role of monotonic sequences. It does not cover related topics such as the importance of natural numbers, geometric progressions, converting sequences to denominator form, bounded sequences, and finding limits of a function. Geometric Progression Introduction to Geometric Progression In mathematics, a geometric progression (G.P) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio. A G.P is an sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number. This fixed number is called the ratio of the progression. Examples of Geometric Progression Here are some examples of geometric progression: 1. 1, 2, 4, 8, 16,... (where the ratio is 2) 2. 1, 1/2, 1/4, 1/8, 1/16,... (where the ratio is 1/2) 3. 3, 6, 12, 24, 48,... (where the ratio is 2) Converting to Denominator Form (Note: Not related to Geometric Progression) Monotonic Decreasing Sequences A monotonic decreasing sequence is a sequence of numbers in which each term is smaller than or equal to its predecessor. Sequence Convergence Theorems (Note: Not related to Geometric Progression) Understanding Sequence Expansion (Note: Not related to Geometric Progression) Monotonic Increasing Sequences A monotonic increasing sequence is a sequence of numbers in which each term is greater than or equal to its predecessor. Calculating Limits to Infinity (Note: Not related to Geometric Progression) Convergence of Sequences (Note: Not related to Geometric Progression) Bounded Sequences (Note: Not related to Geometric Progression) Limit of a Sequence (Note: Not related to Geometric Progression) Finding Limits of a Function (Note: Not related to Geometric Progression) Concept of Monotonic Increasing (Note: Not related to Geometric Progression) Limitations of Infinity in Math (Note: Not related to Geometric Progression) Bounded Sequences In mathematics, a sequence is considered bounded if there exists a real number M such that the absolute value of every term in the sequence is less than or equal to M. This means that the sequence cannot grow without bound in either the positive or negative direction. There are two types of bounded sequences: 1. Bounded Above: A sequence is bounded above if there exists a real number M such that every term in the sequence is less than or equal to M. 2. Bounded Below: A sequence is bounded below if there exists a real number M such that every term in the sequence is greater than or equal to M. A sequence that is both bounded above and bounded below is said to be bounded. Example The sequence (1/n) where n is a natural number is a bounded sequence. It is bounded below by 0 and bounded above by 1, since 0 < 1/n