Sequences and Series Convergence

Study Notes

The concept of sequences and series is a fundamental part of mathematical analysis, dealing with the study of ordered lists of objects, known as sequences, and the sum of the terms of a sequence, referred to as a series.
Sequences and series are used to model various real-world phenomena, such as population growth, financial transactions, and physical systems.
Understanding convergence of sequences and series is crucial, as it determines whether a sequence or series approaches a finite limit or diverges.

Convergence of a sequence or series refers to the tendency of the sequence or series to approach a finite limit as the number of terms increases.
There are various types of convergence, including pointwise convergence, uniform convergence, and absolute convergence.
Convergence tests are used to determine whether a sequence or series converges, including the nth term test, the ratio test, the root test, and the integral test.

A sequence is a list of objects, called terms, in a specific order, often denoted by {an} where n is a natural number.
A series is the sum of the terms of a sequence, often denoted by ∑an.
Convergence of a sequence or series refers to the existence of a finite limit as the sequence or series approaches infinity.

A sequence {an} converges to a limit L if for every positive real number ε, there exists a natural number N such that for all n > N, |an - L| < ε.
A series ∑an converges to a sum S if the sequence of partial sums {Sn} converges to S.
Convergence of a sequence or series is often determined by tests such as the nth-term test, ratio test, and root test.

A sequence is an ordered list of numbers, denoted as {an} where n is a natural number
A series is the sum of the terms of a sequence, denoted as Σan
Convergence of a sequence means that the sequence approaches a fixed value as it goes to infinity
Convergence of a series means that the sum of the terms of a sequence approaches a fixed value as the number of terms increases