Sequences and Series

Definition

A sequence is an ordered list of numbers $a_1,a_2,a_3,\dots$ A series is the sum of a sequence. The partial sums $s_n={\sum }_{k=1}^n{a}_k$ define convergence: an infinite series ${\sum }_{k=1}^{\infty }{a}_k$ converges if ${\lim }_{n\to \infty }{s}_n$ exists and is finite; otherwise it diverges.

Intuition

A series converges when the running total “settles down” to a fixed value. Adding infinitely many positive terms can still yield a finite result if the terms shrink fast enough — but shrinking to zero is necessary, not sufficient.

Formal Description

A sequence may be defined explicitly (e.g. $a_n=1/n$) or by a recursion relation. The Fibonacci sequence is defined by

$$F_{n+2}=F_{n+1}+F_n,F_1=F_2=1.$$

The partial sums $s_1,s_2,\dots ,s_n$ are defined as

$$s_1=a_1,s_2=a_1+a_2,\dots ,s_n=a_1+a_2+\cdots +a_n.$$

The partial sums of the Fibonacci sequence telescope:

$$\sum _{k=1}^n{F}_k=F_{n+2}-1.$$

Key Results

• A series converges if and only if its sequence of partial sums converges.
• Convergence depends only on the tail of the series (finitely many terms do not affect convergence).

Applications

Sequences and series appear throughout analysis, probability (expected values), and discrete mathematics. They are the foundation for Power Series and Taylor Series.

Trade-offs

The definition of convergence via partial sums is precise but gives no direct method for finding the sum. Separate convergence tests (Ratio Test, integral test, comparison) are needed in practice.

Links

• Geometric Series
• Harmonic and p-Series
• Ratio Test
• Power Series