1.1 Sequences: Convergence and Monotony
1.2 Series: Definition and Convergence
1.3 Specific Sequences and Series
Q1. What is meant by a mathematical sequence, their properties, and how to analyze them.
Q2. How the convergence of a sequence is defined and how to check it?
Q3. What are the differences between arithmetical and geometrical sequences?
Q4. How series are formed from sequences, the properties of series, and when a series converges.
Q5. What kind of questions can be modeled with sequences and series?
Q6. Which limit values, which are important for analysis, are found in sequences and series.
NOTES:
Sequence: An enumerated collection of real numbers a1, a2,...,an is called a sequence (an).
Sequence Element: Every single number in a sequence is called a sequence element or simply a member.
Each sequence element is numbered with a natural number n ∈ ℕ, the index. This results in a clear assignment of a natural number n ∈ ℕ to the real sequence element an.
(1)The order in which the sequence elements appear is thus fixed and cannot be changed without changing the sequence itself.
(2)The same element can occur several times in a sequence.
These two characteristics distinguish a sequence from a classical set of numbers.
There are two ways to specify a sequence:
(1) Either you can list all sequence members by writing them one after the other, each separated by a comma, or
(2) you can enter the formula for a general member in the form of an equation.
For example, you can specify the sequence of even numbers by explicitly listing them with (bn) = 2, 4, 6, 8, ...
or using the formula bn = 2.n ∀ n ∈ N.
(∀ = for all) and (∈ = is in / is an element of / belongs to / belonging to).
The formula is read: b sub n is equal to 2 times n for all n belonging to (the natural number) N. ... or n in N.
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