zuai-logo

What does the Alternating Series Test state?

If ana_n is decreasing and limnan=0\lim_{n \to \infty} a_n = 0, then the alternating series (1)nan\sum (-1)^n a_n converges.

All Flashcards

What does the Alternating Series Test state?
If $a_n$ is decreasing and $\lim_{n \to \infty} a_n = 0$, then the alternating series $\sum (-1)^n a_n$ converges.
What does the Ratio Test state?
If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$, then the series converges if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$.
What does the nth Term Test for Divergence state?
If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.
Define a sequence.
A function whose domain is the set of natural numbers.
Define a series.
The sum of the terms of a sequence.
What is a convergent sequence/series?
Terms approach a specific value (the limit).
What is a divergent sequence/series?
Terms do not approach a specific value; the sum is infinite.
Define an arithmetic sequence.
Sequence with a constant difference between consecutive terms.
Define a geometric sequence.
Sequence where each term is the previous term multiplied by a constant ratio.
Define a harmonic series.
Series where terms are reciprocals of positive integers.
Define a power series.
Series of the form $\sum a_n(x-c)^n$ where $a_n$ are constants and $c$ is a constant.
Define an alternating series.
Series where the terms alternate in sign.
Define a Taylor series.
Representation of a function as an infinite sum of terms, with each term being a polynomial function of a single variable.
Define a Maclaurin series.
A Taylor series centered at 0.
What are the differences between Limit Comparison Test and Direct Comparison Test?
Direct Comparison: Compares magnitude directly. Limit Comparison: Compares the limit of the ratio of terms. Direct comparison requires establishing an inequality, limit comparison does not.
What are the differences between Taylor and Maclaurin series?
Taylor: Expansion around any point c. Maclaurin: Expansion around c=0. Maclaurin is a special case of Taylor.
What are the differences between Alternating Series Error Bound and Lagrange Error Bound?
Alternating Series: Applies only to alternating series, uses the next term. Lagrange: Applies to Taylor polynomials, uses the (n+1)th derivative.
What are the differences between arithmetic and geometric sequences?
Arithmetic: Constant difference between terms. Geometric: Constant ratio between terms. Arithmetic: Linear growth. Geometric: Exponential growth.