We now consider some examples of sequences ${z}_{n}$ and series of complex numbers with partial sums ${S}_{n}=\sum _{m=0}^{n}{z}_{m}$.

Their simulation and visualization in the complex plane provides a deeper understanding of the arithmetic operations. It shows a wealth of surprising as well as aesthetically pleasing phenomena, whose study leads to an improved understanding of the underlying mathematical questions. The examples for real series considered above are special cases of similar complex sequences.

A sequence is convergent, if and only if it possesses one accumulation point; an accumulation point is defined such, that an arbitrarily small vicinity of the accumulation point, the accumulation interval , contains in the limit nearly all terms of the sequence.

For the sequences of real numbers that have been discussed above the accumulation point is with respect to the one dimensional domain of the ${F}_{n}$ or ${S}_{n}$. For the geometric sequence or series with the parameter $\left|a\right|<1$, the accumulation point of the sequence is zero and the accumulation point of the series is the real numbers $1\u2215\left(1-a\right)$

The concept of an accumulation point is especially descriptive for complex numbers, since it is enclosed by a small circle in the complex plane.

As for the visualization of the elementary complex operations we use two windows, of which the left shows the terms of the sequence ${z}_{n}$ and the right shows the partial sums ${S}_{n}$ of the series. The unit circle is marked red in both. In the left window the point ${z}_{1}$ (second point of the sequence) corresponding to $a$ is shown enlarged. It can be pulled with the mouse, such that $a$ can be easily changed in this way.

In the right window the first term of the sequence is drawn enlarged; an accumulation point, if present, is encircled by a small green circle.

Remember, that complex multiplication changes not only the absolute value but also the angle, if the imaginary part is not zero. The same thing happens when adding the the terms of the sequence. In general, sequence and series therefore develop on spiral trajectories on the complex plane.

The description in the text can be kept short, since the simulation includes a description window with several pages, of which one is contains instructions for systematic experiments.

The simulation calculates 1000 terms of the sequence. For strong convergence many points coincide close to the accumulation point, such that only a few points can be seen separately on the screen.

The terms of the complex geometric sequence are created in analogy to the real case with the rule:

$$\begin{array}{c}{z}_{0}=1\hfill \\ {z}_{n+1}={z}_{n}\cdot a;\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}n\ge 0\to {z}_{n}={a}^{n}\hfill \\ \hfill \end{array}$$

Here ${z}_{n}$ is the $n$-th term of the sequence. The parameter $a$ is a complex number. The terms are thus given by $1,a,{a}^{2},{a}^{3},{a}^{4},\dots $; the first term ${z}_{0}$ is also equal to one independent of $a$.

The complex geometric series is created via continuous addition of the terms of the complex sequence. Its partial sums are:

$${S}_{n}=\sum _{m=0}^{n}{a}^{m}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}};\text{}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{S}_{n\text{}}=1+{a}^{1}+{a}^{2}+\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{a}^{3}....+\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{a}^{n}\text{}$$

The first partial sum ($n=0$) is again independent of $a$ always one.

In the simulation shown in Fig.4.3 you can move the point $a$ (the second point in the sequence) in the left complex plane with the mouse and observe the effect on the terms of the sequence on the left hand plane and on the partial sums of the complex series on the right hand plane.

The simulation is started via the Ctrl-key and clicking on the following diagram. The complex geometric series converges, if the absolute value of $a$ is smaller than $1$, i.e. if $a$ lies inside of the thin red unit circle that is drawn in the left hand plane.

In the case of convergence the limit of the series is:

$$\underset{n\to \infty}{lim}{S}_{n}=\underset{n\to \infty}{lim}\sum _{m=0}^{n}{a}^{m}=\frac{1}{1-a}$$

It is situated in the center of the green accumulation circle drawn in the right window.

For $\left|a\right|>1$ the series diverges. The unit circle becomes smaller and smaller in the growing domain of coordinates and the series runs away along a spiral to infinity.

The case of the real geometric series is obtained as special case of the complex series, if the point $a$ is moved along the real axis. To look at the situation in more detail you can maximize the simulation window to full screen size. On the inner boundary of the unit circle the convergence can be so slow, that 1000 terms are not sufficient to nearly reach the limit. This can lead to very interesting geometrical patterns.

The terms of the complex exponential sequence are created with the following rule:

$$\begin{array}{c}\text{exponentialsequence}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{z}_{n+1}={z}_{n}\cdot \frac{a}{n}\hfill \\ \text{geometricsequenceforcomparison:}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{z}_{n+1}={z}_{n}\cdot a\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\hfill \\ \hfill \end{array}$$

Here ${z}_{n}$ is the $n$-th term of the sequence. The parameter $a$ can be a complex number. We again have ${z}_{0}=1$

The terms thus have the form:

$$\begin{array}{c}1,\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\frac{a}{1}\text{},\text{}\frac{a{\text{}}^{2}}{1\cdot 2},\text{}\frac{{a}^{3}}{1\cdot 2\cdot 3},\text{}\frac{{a}^{4}}{1\cdot 2\cdot 3\cdot 4}....\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{z}_{n}\text{}=\frac{{a}^{n}}{n!}\hfill \\ \text{n-}\text{factorial:}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}n!=1\cdot 2\cdot 3\cdot 4...\cdot n;\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}0!\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}=1\phantom{\rule{0em}{0ex}};\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}1!=1\hfill \\ \hfill \end{array}$$

The complex exponential series is created via continued addition of the terms of the complex exponential sequence. Thus its partial sums are:

$$\begin{array}{c}{S}_{n}=\text{}\sum _{m=o}^{n}\frac{{a}^{m}}{m!}\hfill \\ {S}_{n}=\frac{1}{0!}+\frac{a}{1!}+\frac{{a}^{2}}{2!}....+\frac{{a}^{n}}{n!}=1+a+\frac{{a}^{2}}{2}....+\frac{{a}^{n}}{n!}\hfill \\ {S}_{0}=1\hfill \\ \hfill \end{array}$$

The complex sequence and series are shown in Figure 4.4.

The case of the real exponential series is obtained as special case of the complex series, if the point $a$ is chosen on the real axis.

The terms of the exponential sequence always converge to zero. The exponential sequence converges for every finite value of $a$. The convergence is so fast, that the simulation window will only show a few of the 1000 calculated terms separately.

Why does the exponential series converge so quickly and in general as compared to the geometric series? In order to understand this we again consider the ratio of consecutive terms of both sequences:

$$\begin{array}{c}\text{geometricsequence}\frac{{z}_{n+1}}{{z}_{n}}=a\hfill \\ \text{exponentialsequence}\frac{{z}_{n+1}}{{z}_{n}}=\frac{a}{n}\hfill \\ \hfill \end{array}$$

For the geometric series we must have $a<1$, in order for the terms of the sequence to decrease, and this applies to all terms. For the exponential series , the initial terms of the series can even grow strongly! Irrespective of the size of $\left|a\right|$. there is always an index $n$ from which the terms get smaller and smaller in absolute value, independent of the chosen $a$-value. There fore we have ${z}_{n}\to 0$ irrespective of the chosen value of $a$.

One can easily generalize the statement concerning the convergence of the exponential series: We are given a bounded sequence ${B}_{n}$ of numbers, that are multiplied with the respective terms of the exponential sequence. The new series is thus given by:

$$\begin{array}{c}S=\text{}\sum _{m=0}^{\infty}{B}_{m}\frac{{a}^{m}}{m!}={B}_{0}+{B}_{1}a+{B}_{2}\frac{{a}^{2}}{1\cdot 2}...+{B}_{m}\frac{{a}^{m}}{m!}+...\hfill \\ \left|{B}_{m}\right|\phantom{\rule{0em}{0ex}}q\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{with}q\text{real,positivenumber}\to \hfill \\ \phantom{\rule{0em}{0ex}}\left|S\right|\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}q\sum _{m=0}^{\infty}\frac{{a}^{m}}{m!}\phantom{\rule{0em}{0ex}};\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}S\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{is}\phantom{\rule{0em}{0ex}}\text{convergent,}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{since}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\sum _{m=0}^{\infty}\frac{{a}^{m}}{m!}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{converges}\hfill \\ \hfill \end{array}$$

If the absolute values of the coefficients ${B}_{m}$ stay smaller than an arbitrary large real number $q$ , i.e. the sequence ${B}_{m}$ does not diverge, then the series converges, since it is smaller as the convergent exponential function multiplied by a real number. This shows, how strongly the exponential series itself converges. We will later apply this result to the convergence of the Taylor expansion.

For the limit of the exponential series we have:

$$\begin{array}{c}\underset{n\to \infty}{lim}{S}_{n}=\underset{n\to \infty}{lim}\sum _{m=0}^{n}\frac{{a}^{m}}{m!}={e}^{a};\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}e=2.71828\dots \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{Euler\u2019s}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{Number}\hfill \\ \text{If}a=1\text{oneobtains}\hfill \\ e=\underset{n\to \infty}{lim}\sum _{m=0}^{n}\frac{1}{m!}=1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}...\hfill \\ \hfill \end{array}$$

If one moves $a$ in the simulation along the imaginary axis, the limit of the series moves on a circle around the origin. Thus one obtains “experimentally” the famous Euler formula

$$\begin{array}{c}\text{With}a=x+iy\hfill \\ {e}^{a}={e}^{x}{e}^{iy}={e}^{x}\left(cosy+isiny\right)\hfill \\ \text{For}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}x=0\to {e}^{iy}=cosy+isiny\hfill \\ {e}^{iy}=1+iy-\frac{{y}^{2}}{2!}-\phantom{\rule{0em}{0ex}}i\frac{{y}^{3}}{3!}+\frac{{y}^{4}}{4!}+\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}i\frac{{y}^{5}}{5!}...\phantom{\rule{0em}{0ex}}=1-\frac{{y}^{2}}{2!}\phantom{\rule{0em}{0ex}}+\frac{{y}^{4}}{4!}\phantom{\rule{0em}{0ex}}-...\phantom{\rule{0em}{0ex}}+i\left(y-\frac{{y}^{3}}{3!}+\frac{{y}^{5}}{5!}...\right)\hfill \\ \to \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}cosy=\sum _{n=0}^{\infty}{\left(-1\right)}^{n}\frac{{y}^{2n}}{\left(2n\right)!};\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}siny=\sum _{n=0}^{\infty}{\left(-1\right)}^{n}\frac{{y}^{2n+1}}{\left(2n+1\right)!}\hfill \\ \hfill \end{array}$$

Eulers Formula is useful for the easy derivation of relationships involving trigonometric functions. Two examples:

$$\begin{array}{c}\text{wearelookingfor}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}cos2\varphi ,\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}sin2\varphi \hfill \\ cos2\varphi +i\phantom{\rule{0em}{0ex}}sin2\varphi ={e}^{i2\varphi}={\left({e}^{i\varphi}\right)}^{2}\to \hfill \\ cos2\varphi +i\phantom{\rule{0em}{0ex}}sin2\varphi =\left(cos\varphi +i\phantom{\rule{0em}{0ex}}sin\varphi \right)\left(cos\varphi +i\phantom{\rule{0em}{0ex}}sin\varphi \right)={\left(cos\varphi \right)}^{2}-{\left(sin\varphi \right)}^{2}+i2cos\varphi sin\varphi \hfill \\ \to cos2\varphi ={\left(cos\varphi \right)}^{2}-{\left(sin\varphi \right)}^{2}\hfill \\ \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}sin2\varphi =2cos\varphi sin\varphi \hfill \\ \text{wewouldliketoevaluate}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}cos\left({\varphi}_{1}+{\varphi}_{2}\right),\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}sin\left({\varphi}_{1}+{\varphi}_{2}\right)\hfill \\ cos\left({\varphi}_{1}+{\varphi}_{2}\right)+i\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}sin\left({\varphi}_{1}+{\varphi}_{2}\right)={e}^{i\left({\varphi}_{1}+{\varphi}_{2}\right)}={e}^{i{\varphi}_{1}}{e}^{i{\varphi}_{2}}=\left(cos{\varphi}_{1}+i\phantom{\rule{0em}{0ex}}sin{\varphi}_{1}\right)\left(cos{\varphi}_{2}+i\phantom{\rule{0em}{0ex}}sin{\varphi}_{2}\right)\hfill \\ \to cos\left({\varphi}_{1}+{\varphi}_{2}\right)=cos{\varphi}_{1}cos{\varphi}_{2}-sin{\varphi}_{1}sin{\varphi}_{2}\hfill \\ \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}sin\left({\varphi}_{1}+{\varphi}_{2}\right)=cos{\varphi}_{1}\phantom{\rule{0em}{0ex}}sin{\varphi}_{2}+sin{\varphi}_{1}cos{\varphi}_{2}\hfill \\ \hfill \end{array}$$

Whenever one works with oscillations, i.e. with trigonometric functions for example in optics and electronics using complex numbers has many practical advantages.

From Eulers formula we obtain an elegant approximation formula for $\pi $ if we put $y=\pi $ (You may convince yourself in the the simulation that the exponential function indeed yields $-1$ for $z=i\pi $).

$$\begin{array}{c}y=\pi \to {e}^{i\pi}=cos\pi +isin\pi \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}=-1+i\cdot 0\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}=\phantom{\rule{0em}{0ex}}-1\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\hfill \\ {e}^{i\pi}=1+i\pi -\frac{{\pi}^{2}}{2!}-\frac{i{\pi}^{3}}{3!}+\frac{{\pi}^{4}}{4!}+i\frac{{\pi}^{5}}{5!}-\frac{{\pi}^{6}}{6!}-i\frac{{\pi}^{7}}{7!}...\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}=-1\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\hfill \\ \text{Separationinrealandimaginaryparts}\to \hfill \\ \text{Re}\to \text{}2=\frac{{\pi}^{2}}{2!}-\frac{{\pi}^{4}}{4!}+\frac{{\pi}^{6}}{6!}-\frac{{\pi}^{8}}{8!}+\frac{{\pi}^{10}}{10!}-....\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\hfill \\ \text{Im}\to \text{}0=\pi -\frac{{\pi}^{3}}{3!}+\frac{{\pi}^{5}}{5!}-\frac{{\pi}^{7}}{7!}+\frac{{\pi}^{9}}{9!}-\frac{{\pi}^{11}}{11!}+...\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\pi \ne 0\to 0=1-\frac{{\pi}^{2}}{3!}-\frac{{\pi}^{4}}{5!}+\frac{{\pi}^{6}}{7!}-\frac{{\pi}^{8}}{9!}+\frac{{\pi}^{10}}{11!}-....\hfill \\ \hfill \end{array}$$

The equations are polynomials in ${\pi}^{2}$. Neglecting all higher powers the two series yield in zeroth order the solutions $\sqrt[2]{4}=2;\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\sqrt[2]{6}=2.449....$. Using iterative methods of solution , for example fixpoint iteration in EXCEL, one obtains the following quickly converging values, which are listed below together with the highest powers taken into account:

$$\begin{array}{c}\text{approximationsusingthelastequations(inbracketsthehighestpowerof}\pi )\text{kept:}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\hfill \\ \left({\pi}^{2}\right)\to \sqrt{6}=2.4;\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\left({\pi}^{6}\right)\phantom{\rule{0em}{0ex}}\to \phantom{\rule{0em}{0ex}}3.078;\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\left({\pi}^{10}\right)\phantom{\rule{0em}{0ex}}\to \phantom{\rule{0em}{0ex}}3.1411;\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\left({\pi}^{14}\right)\phantom{\rule{0em}{0ex}}\to \phantom{\rule{0em}{0ex}}3.1415920\hfill \\ \hfill \end{array}$$

Subtraction of both equations leads to a series that converges even faster, with the zeroeth order solution for the order $p{i}^{4}$: $\sqrt[4]{60}=2.78$.