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Arithmetic and geometric series

Arithmetic sequences and 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

Geometric sequences

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.

Geometric series

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.

Convergence and monotonicity for 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.