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Series and Sequences
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Arithmetic Sequence
A sequence in which each term after the first is obtained by adding a constant difference. Formula:
Geometric Sequence
A sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Formula:
Harmonic Sequence
A sequence of numbers formed by taking the reciprocals of an arithmetic sequence. No specific formula, but can be expressed as where is the difference between the terms in the underlying arithmetic sequence.
Fibonacci Sequence
A sequence where each term is the sum of the two preceding ones, usually starting with 0 and 1. Formula:
Arithmetic Series
The summation of an arithmetic sequence. Formula for the sum of the first terms:
Geometric Series
The summation of a geometric sequence. Formula for the sum of the first terms: , where
Infinite Geometric Series
A geometric series with infinite terms. Formula for the sum: , where
P-Series
A series of the form , where is a positive constant. Converges when and diverges when .
Telescoping Series
A series where most terms cancel out when summed. Explicit formula varies, but typically has a form where results in many terms canceling out.
Power Series
An infinite series of the form where represents the coefficients and is a constant.
Taylor Series
A type of power series that represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Formula:
Maclaurin Series
A special case of the Taylor series centered at . Formula:
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