Glossary
Alternating Error Bound
A method to estimate the maximum error when approximating the sum of a convergent alternating series using a partial sum. The error is less than or equal to the absolute value of the first unused term.
Example:
If you approximate the sum of the alternating harmonic series using its first 100 terms, the Alternating Error Bound guarantees your error is no larger than the absolute value of the 101st term, 1/101.
Alternating Series
A series in which the terms alternate in sign, typically having a factor of (-1)^n or (-1)^(n+1).
Example:
The series 1 - 1/2 + 1/3 - 1/4 + ... is an alternating series that converges to ln(2).
Alternating Series Test
A test for alternating series: if the absolute value of the terms decreases monotonically and the limit of the terms is zero, then the series converges.
Example:
For the series ∑((-1)^(n+1))/n, the terms 1/n decrease and approach zero, so the Alternating Series Test confirms its convergence.
Arithmetic Sequence
A sequence where the difference between consecutive terms is constant, known as the common difference.
Example:
The sequence 5, 8, 11, 14, ... is an arithmetic sequence with a common difference of 3.
Convergent (Sequence/Series)
A sequence or series is convergent if its terms approach a specific finite value (a limit) as the number of terms approaches infinity, meaning its sum is finite.
Example:
The series 1 + 1/2 + 1/4 + 1/8 + ... is convergent because its sum approaches the finite value of 2.
Direct Comparison Test
A test that compares the terms of two series. If a larger series converges, a smaller series also converges. If a smaller series diverges, a larger series also diverges.
Example:
Since 1/n^2 is always less than or equal to 1/n for n≥1, if you know ∑(1/n) diverges, then ∑(1/(n-1)) for n>1 also diverges by the Direct Comparison Test.
Divergent (Sequence/Series)
A sequence or series is divergent if its terms do not approach a specific finite value, or if the sum of its terms approaches infinity or oscillates without bound.
Example:
The series 1 + 2 + 3 + 4 + ... is divergent because its sum grows infinitely large.
Geometric Sequence
A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Example:
The sequence 3, 6, 12, 24, ... is a geometric sequence with a common ratio of 2.
Harmonic Series
A specific divergent series where the terms are the reciprocals of the positive integers, represented as 1 + 1/2 + 1/3 + 1/4 + ...
Example:
Despite its terms getting smaller, the Harmonic Series is a classic example of a series that diverges, meaning its sum grows infinitely large.
Integral Test
A test that determines the convergence or divergence of a series by comparing it to an improper integral of a related continuous, positive, and decreasing function.
Example:
To determine if the series ∑(1/n^2) converges, you can use the Integral Test by evaluating the improper integral ∫(1/x^2) dx from 1 to infinity.
Interval of Convergence
The set of all x-values for which a power series converges. It is determined by the radius of convergence and by testing the endpoints of the interval.
Example:
If a power series has a radius of convergence of 3 centered at x=1, its interval of convergence might be (-2, 4), [-2, 4), (-2, 4], or [-2, 4] depending on endpoint behavior.
Lagrange Error Bound
A formula used to determine the maximum possible error when approximating a function with its Taylor polynomial. It involves the maximum value of the next derivative of the function on the given interval.
Example:
When using a 3rd-degree Taylor polynomial to estimate cos(0.5), the Lagrange Error Bound helps you quantify the largest possible error in your approximation, ensuring your estimate is within a certain precision.
Limit Comparison Test
A test that compares two series by taking the limit of the ratio of their terms. If the limit is a finite, positive number, then both series either converge or diverge together.
Example:
To determine if ∑(1/(n^2 + 1)) converges, you could use the Limit Comparison Test with the known convergent p-series ∑(1/n^2).
Maclaurin Series
A special case of the Taylor series where the center of expansion is 0. It represents a function as an infinite polynomial based on its derivatives evaluated at zero.
Example:
The Maclaurin Series for cos(x) is 1 - x^2/2! + x^4/4! - ..., which is a simple way to approximate cosine values near zero.
Power Series
A series of the form ∑a_n(x-c)^n, where a_n are constants, c is the center, and x is a variable. It represents a function as an infinite polynomial.
Example:
The series x - x^3/3! + x^5/5! - ... is a power series representation for sin(x), centered at c=0.
Power Series Representation
The expression of a function as an infinite sum of terms involving powers of (x-c), typically derived from a Taylor or Maclaurin series.
Example:
The power series representation for 1/(1-x) is 1 + x + x^2 + x^3 + ..., which is a geometric series.
Radius of Convergence
For a power series centered at c, it is the distance R from c such that the series converges for all x in the interval (c-R, c+R) and diverges for |x-c| > R.
Example:
For the power series ∑(x^n/n!), the radius of convergence is infinity, meaning it converges for all real numbers x.
Ratio Test
A test that determines convergence by finding the limit of the ratio of consecutive terms. If the limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, it's inconclusive.
Example:
To find the radius of convergence for a power series like ∑(x^n/n!), the Ratio Test is typically used.
Sequence
An ordered list of numbers, often defined by a rule or formula, where the domain is the set of natural numbers.
Example:
The list of numbers 1, 3, 5, 7, ... is an arithmetic sequence where each term is found by adding 2 to the previous one.
Series
The sum of the terms of a sequence. It can be finite or infinite.
Example:
The expression 1 + 1/2 + 1/4 + 1/8 + ... represents a geometric series where each term is half of the previous one.
Series Analysis
The study of functions and their behavior over time, specifically focusing on the properties and convergence/divergence of infinite series.
Example:
When a physicist models the long-term behavior of a decaying radioactive substance, they might use series analysis to understand if the total decay approaches a finite value.
Taylor Series
A representation of a function as an infinite sum of terms, calculated from the values of the function's derivatives at a single point (the center).
Example:
To approximate a complex function like e^x around x=2, you would use a Taylor Series centered at 2, building a polynomial from its derivatives at that point.
nth Term Test/Limit Test
A test for divergence: if the limit of the nth term of a series as n approaches infinity is not zero, then the series diverges. If the limit is zero, the test is inconclusive.
Example:
For the series ∑(n/(n+1)), the limit of the nth term is 1 (not zero), so the nth Term Test tells us the series diverges.