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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.

Arithmetic sequences and their corresponding series
Arithmetic sequences and their corresponding series

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 early growth of a bacterial colony is a geometric sequence.
The early growth of a bacterial colony is a geometric sequence.

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 early growth of a bacterial colony is a geometric sequence. If the bacteria do not die, the total number is a geometric series.
The early growth of a bacterial colony is a geometric sequence. If the bacteria do not die, the total number is a geometric series.

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).
The sequence has a ratio of $$1/2$$, so the sequence and the series converge.
The sequence has a ratio of $$1/2$$, so the sequence and the series converge.