Geometrical series

The geometrical series is one of the simplest examples of numerical Série that one can give. It is the series of general term $\ lambda^n$ where $\ lambda$ is a real number (level first) or complex (level first academic year).

Although seemingly simple, it deserves attention because it admits a generalization in the algebras of Banach which makes it possible to study the variations of the reverse of an element.

What a geometrical series?

The geometrical series of initial term has and of geometrical ratio Q (real different from 1) is the series of general term $a.q^n$ . Its partial sums are calculated explicitly:

S_n=a \ sum_ {i=1} ^ {N} q^ {i-1} =a. \ frac {1-q^ {N}} {1-q}

There exist several demonstrations of this formula:

Proof by recurrence: The identity is true for $n=1$ . Let us suppose checked it with the row N . Then, it is enough to write:

S_ {n+1} =S_n+a.q^n=a. \ frac {1-q^ {N}} {1-q} +a.q^n=a. \ frac {1-q^ {N} +q^n-q^ {n+1}} {1-q} =a. \ frac {1-q^ {n+1}} {1-q}

more astute Proof: One multiplies $S_n$ by $q$ , then one withdraws $S_n$ from the result obtained.

\begin{array}{ccccccccccccccc} S_n &=& has &+& a.q &+& aq^2 &+& \ ldots &+& aq^ {N2} &+& aq^ {n-1} & & \ \ qS_n &=& & & a.q &+& aq^2 &+& \ ldots &+& aq^ {N2} &+& aq^ {n-1} &+& aq^n \ \ \ hline S_n-qS_n &=& has & & & & & & & & & & &-& aq^n \ \ \ end {array}
One obtains

S_n \ left (1-q \ right) =a \ left (1-q^n \ right)

then

S_n=a \ cfrac {1-q^n} {1-q}

This geometrical series is convergent if and only if the continuation $(S_n)$ converges. In this study, one distinguishes three cases (by eliminating the case a=0 which are without interest):

When $|Q|<1$ , in this case, $q^n$ tends towards 0, and thus the continuation $(S_n)$ is convergent, of limit $\ cfrac {has} {1-q}$ .

It should be noted that this concept makes it possible to solve the Paradoxe of Achilles and the tortoise stated by the old Greeks.

When $|Q|=1$ : if q=1, Then $S_n=a \ cdot n$ ; if $q=-1$ , then $S_n=0$ for $n$ even and $S_n = a$ for $n$ odd. The series diverges in both cases.
When $|Q|>1$ , the continuation $(q^n)$ and a fortiori $(S_n)$ diverges coarsely.

These sums are known as geometrical, because they appear by comparing lengths, surfaces, volumes, geometrical etc of forms in various dimensions. The result not quite difficult but in any case remarkable is the following:

The geometrical series of initial term has not no one and of reason Q is convergent if and only if the absolute Value of the reason is strictly lower than 1 ; id is:

|Q|< 1

. In this case, its sum is worth:

\ sum_ {n=0} ^ {\ infty} a.q^n= \ frac {has} {1-q}

Numerical example

2 + 4 + 8 + 16 + 32 + 64 + 128 + 256

It is the sum partial of a geometrical series of report/ratio 2 and first term 2. The formula above is written:

2 + 4 + 8 + 16 + 32 + 64 + 128 + 256=2 \ times \ frac {1-256} {1-2} =510

1/3 = 0.33333…. can conceive itself like the limit of a geometrical series of reason 1/10. It is amusing to see that its triple (value 1) can as well be written 1 as 0.9999999…

Generalization with the body of the complexes

The results extend very naturally to the body from the complex numbers.

A geometrical series of first term has and of reason Q , complex numbers, of general term $a.q^n$ is the series. The requirement and sufficient of convergence is that the reason Q is a complex of module strictly lower than 1.

To note that the geometrical series are the simplest examples of whole series which one lays out. Its ray of convergence is 1.

Item 1 is a point of caesura.

Geometrical series in the algebras of Banach

If $(has, \|. \|)$ indicates an algebra of Banach, the series geometry of reason $u \ in A$ is the series of general term U . When $\|U \|<1$ , under-multiplicativité gives:

\|u^n \|\ Leq \|U \|^n

Like the geometrical series real of reason

\|U \|

is convergent, the geometrical series of reason U is absolutely convergent. Let us note S its sum. Then one a:

(1-u). S=\sum_{n=0}^{\infty} u^n-\sum_{n=1}^{\infty} u^n=1

Thus S is the reverse of

(1-u)

. What can seem to be a school exercise of the first cycle proves to be fundamental. Here some applications stated without demonstration:

the whole of the invertible elements of has is open.
For an element X of has , its spectrum - the whole of the complexes $\ lambda$ such as $(\ lambda-x)$ is not invertible - is a nonempty and limited closed part C .
On its field of definition, the application $\ lambda \ mapsto (\ lambda-x) ^ {- 1}$ is developable in whole series.

References

DELHEZ, Eric J. - Mr., Analyzes Mathematical, Tome II, Université of Liege, Belgium, July 2005, p. 344.

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