MA141 Analysis 1

This flashcard set covers vocabulary and key concepts from the MA141 Analysis 1 lecture notes related to sequences, series, continuity, and convergence.

Cauchy sequence

A sequence is a Cauchy sequence if for all ε > 0, there exists N such that for all n, m ≥ N, |xn - xm| < ε.

Bolzano-Weierstrass theorem

Every bounded sequence has a convergent subsequence.

Continuous function

A function f is continuous at point a if for every ε > 0, there exists δ > 0 such that for every x with |x - a| < δ, |f(x) - f(a)| < ε.

Convergence of a series

A series ∑ak converges if the sequence of partial sums Sn converges to a limit A.

Limit superior (lim sup)

The limit sup of a sequence is the largest value that the sequence gets arbitrarily close to infinitely often.

Limit inferior (lim inf)

The limit inf of a sequence is the smallest value that the sequence gets arbitrarily close to infinitely often.

The Intermediate Value Theorem (IVT)

If f is continuous on [a, b] and f(a) < f(b), then for every y in (f(a), f(b)), there is a c in (a, b) such that f(c) = y.

Absolute convergence

A series ∑ak converges absolutely if ∑|ak| converges.

Ratio Test

Convergence criteria for series: If lim |ak+1/ak| = L, then if L < 1, the series converges absolutely; L > 1 means it diverges.

Root Test

If lim |ak|^(1/k) = L, then if L < 1, the series converges absolutely; L > 1 means it diverges.

Increasing sequence

A sequence (xn) is increasing if xn+1 ≥ xn for all n.

Decreasing sequence

A sequence (xn) is decreasing if xn+1 ≤ xn for all n.

Continuous on an interval

A function is continuous on an interval I if it is continuous at every point in I.

Extrema in a continuous function

According to the Extreme Value Theorem, a continuous function on a closed bounded interval attains its minimum and maximum.

Dedekind cuts

A method of constructing real numbers by partitioning the rationals into two non-empty sets with specific properties.

Rational numbers

Numbers that can be expressed as the quotient or fraction p/q of two integers.

Real numbers

The set of numbers that includes rational and irrational numbers.

Supremum and Infimum

Supremum is the least upper bound of a set, while Infimum is the greatest lower bound.

Archimedean property

For any real number x, there exists a natural number n such that n > x.

Existence of roots

There exists a unique non-negative root η such that η^k = a for a ≥ 0 and k in Natural numbers.

Geometric series

A series of the form ∑x^k converges to 1/(1-x) if |x| < 1.

Comparison Test

A method to determine convergence of series based on comparison with another series.

Conditionally convergent series

A series that is convergent but not absolutely convergent.

Images of intervals

If f is continuous, then f(I) is also an interval if I is an interval.

Inverse function theorem

If f is a strictly increasing continuous function, its inverse is also continuous and strictly increasing.

Continuous functions from continuous functions

The sum, product, and composition of continuous functions are continuous.

Convergence of a sequence

A sequence (xn) converges to x if for all ε > 0, there exists N such that |xn - x| < ε for all n ≥ N.