# Arithmetic and geometric series

An arithmetic sequence is a sequence for which the difference of two successive terms is constant: $$$u_{n+1}-u_n = d.$$$The difference $$d$$ is called the common difference.

The general term of an arithmetic sequence is of the form $$$ u_n = u_1 + (n-1) d.$$$ Arithmetic sequences are often called **arithmetic progressions**.

The general term of an ** arithmetic series ** is $$$ S_n= \sum_{k=1}^n u_k = nu_1 + d\frac{n(n-1)}{2} = \frac{n(u_1+u_n)}{2}.$$$

If $$d\gt0$$ (resp. $$d\lt0$$), the sequence is increasing (resp. decreasing) and its limit is $$+\infty$$ (resp. $$-\infty$$). If $$d=0$$, the sequence is constant.

The form of an arithmetic sequence / series is found by induction.

Geometric sequences and series represent quantities with multiplicative growth, such as world population or re-invested bank interests.

A geometric sequence with common ratio $$r$$ is a sequence for which the ratio of two successive terms is constant, i.e. $$$u_{n+1} = ru_n.$$$

The general term of a **geometric sequence** is: $$$ u_n = u_1r^{n-1}$$$

The general term of a **geometric series** with ratio $$r\ne 1$$ is: $$$ S_n= \sum_{k=1}^n u_k = u_1\frac{r^n-1}{r-1}$$$ For $$r=1$$, the series is $$S_n = nu_1$$, because the sequence is constant.

For $$\vert r\vert\lt 1$$, the series converges and is $$$ \sum_{k=1}^{\infty} u_k = \frac{u_1}{1-r}. $$$

The form of a geometric sequence / series can be proved by induction. For the series, it can be seen as well from the identity $$$ (1+r+\dots+r^{n-1}) (r-1)= r - 1 + r^2 - r +\dots r^{n} - r^{n-1} = r^n-1$$$ after dividing it by $$(r-1)$$ when $$r\neq 1$$.

The convergence of geometric sequences and series is summarised below.

- If $$\vert r\vert\lt 1$$, the sequence converges to $$0$$. The series also converges.
- If $$\vert r\vert\gt 1$$, the sequence diverges (i.e. doesn't converge). The series diverges.
- If $$r=1$$, the sequence is constant. It converges. The series diverges unless $$u_1=0$$.
- If $$r=-1$$, the sequence alternates and is of the form: $$u_1$$, $$-u_1$$, $$u_1$$, $$\dots$$ The sequence and the series diverge unless $$u_1=0$$.

The monotonicity properties of geometric sequences are given below.

- If $$r \gt 1$$ and $$u_1 \gt 0$$ (resp. $$u_1\lt 0$$), the geometric sequence is increasing (resp. decreasing). Its limit is $$+\infty$$ (resp. $$-\infty$$).
- If $$0 \lt r \lt 1$$ and $$u_1 \gt 0$$ (resp. $$u_1\lt 0$$), the geometric sequence is decreasing (resp. increasing). Its limit is $$0$$.
- If $$r \lt 0$$, the sequence alternates (each successive term changes sign).