Sequence and series

Sequences are infinite lists of numbers written in a defined order, ascending or descending following a specific rule. Series are the sum of all the terms in a sequence, either finite or infinite.

Arithmetic and geometric

An arithmetic sequence grows with a linear relationship, i.e., through addition. We find the $n$th term of such a sequence, where $u_i$ denotes the $i$th term and $d$ the common difference, with:

$$u_n=u_1+(n-1)d$$

And the sum of $n$ terms of an arithmetic sequence with:

$$S_n=\frac{n}{2}(2u_1+(n-1)d)=\frac{n}{2}(u_1+u_n)$$

A geometric sequence grows with a constant ratio between terms, i.e., through multiplication. We denote $r$ the common ratio, and find the $n$th term with:

$$u_n=u_1{r}^{n-1}$$

For the sum of $n$ terms of a finite geometric sequence, where $r\ne 1$:

$$S_n=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r}$$

Curiously, at times we have a finite sum for an infinite geometric sequence. We say such a sequence is convergent when there’s a common ratio $r<1$, and divergent when it doesn’t converge to some value.

$$S_{\infty }=\frac{u_1}{1-r}$$

Extensions

Whether series converge or diverge is an important discussion of its own right. An important extension on series are power series, which are able to approximate functions with polynomials.

• Series convergence
• Power series
  • Taylor series
  • Laurent series
• Fourier series
  • Fourier series convergence
  • Continuous-time Fourier series
  • Discrete-time Fourier series
• Cauchy sequence