The study of sequences and series in the context of calculus.
Arithmetic Sequences: A sequence in which the difference between consecutive terms is constant.
Geometric Sequences: A sequence in which the ratio between consecutive terms is constant.
Convergent and Divergent Sequences: A convergent sequence approaches a finite limit as the number of terms approaches infinity, while a divergent sequence has no limit.
Partial Sums: The sum of a finite number of terms in a sequence is called the partial sum of that sequence.
Infinite Series: An infinite series is the sum of an infinite number of terms in a sequence.
Tests for Convergence and Divergence: There are various tests to determine whether an infinite series converges or diverges, such as the ratio test and the root test.
Power Series: A power series is an infinite series where each term is a power of a variable multiplied by a coefficient.
Taylor Series: A Taylor series is a power series that approximates a function by using the values of its derivatives at a single point.
Maclaurin Series: A Taylor series at x=0 is called a Maclaurin series.
Fourier Series: A Fourier series is a series that represents a periodic function as a sum of sines and cosines.
Absolute Convergence: An infinite series is said to converge absolutely if the sum of the absolute values of its terms is finite.
Alternating Series: An alternating series is a series in which the signs of the terms alternate.
Ratio Test: The ratio test is a method used to determine the convergence or divergence of an infinite series.
Root Test: The root test is a method used to determine the convergence or divergence of an infinite series based on the growth rate of its terms.
Binomial Series: A binomial series is a special type of power series that has a finite number of terms.
Zeta Function: The zeta function is a mathematical function that can be used to study the distribution of prime numbers.
Exponential Function: The exponential function is a function that grows exponentially as the variable increases.
Trigonometric Functions: The trigonometric functions are mathematical functions that relate the angles of a right triangle to the lengths of its sides.
Complex Numbers: Complex numbers are numbers that have both a real and an imaginary part.
Vector Calculus: Vector calculus is a branch of calculus that deals with vector fields, which are functions that assign a vector to each point in space.
Arithmetic sequence: A sequence in which each term is obtained by adding a fixed number to the previous term.
Geometric sequence: A sequence in which each term is obtained by multiplying the previous term by a fixed number.
Harmonic sequence: A sequence in which each term is the reciprocal of a natural number.
Fibonacci sequence: A sequence in which each term is the sum of the two preceding terms, starting with 0 and 1.
Binomial series: A series that arises from expanding (1+x)^n, where n is a non-negative integer.
Power series: A series in which each term is a constant times a variable raised to a power.
Alternating series: A series in which the signs of the terms alternate between positive and negative.
Convergent series: A series with a finite limit.
Divergent series: A series that does not have a finite limit.
Telescoping series: A series that can be expressed as the difference between two series.
Taylor series: A power series that represents a function as an infinite sum of its derivatives.
Maclaurin series: The Taylor series of a function at x=0.
Fourier series: A series that represents a periodic function as an infinite sum of sines and cosines.
Zeta function: A series that arises from the infinite sum of the reciprocals of the natural numbers raised to a power.