# If n + 1 n what is n

A __Sequence of numbers__ is with a_{n} designated and their __Sequential members__ obey that __Education Act__ the sequence of numbers.

Natural numbers (sometimes also with 0) are used for n.

For example: a_{n} = n + 2 (n + 2 is the law of formation; values for n = ... are elements of the sequence)

or: a_{n} = (½)^{n}

or also: a_{n} = (-1)^{n} (n / 2) + 1

While in the second example the sequence values for larger n are getting smaller and smaller, in the third example you can see that the values for even n are getting larger and smaller for odd n.

A sequence of numbers can also __recursive__ can be defined as the following example shows:

The first seven terms of a sequence are shown here, in which an ever higher power of -1/2 is added and subtracted alternately.

The consequence obeys the following law:

a_{n + 1} = a_{n} + (-½)^{ n} and a_{0} = 0

This is called a recursively (going backwards) defined sequence, since a sequence term can only be calculated if you know its predecessor.

In the figure below, the graph is the sequence **a _{n}= 1 (1 / n)** shown:

This consequence is __increasing monotonously__because each sequence value is greater than its predecessor. This can be shown by proving: a_{n + 1} a_{n}> 0.

There is also analog __monotonically falling__ Follow like a_{n} = 1 + (1 / n). (Proof by: a_{n + 1} a_{n}< 0.)

If you look at the above illustration, you will notice that the values are getting closer and closer to 1. For example, a_{4 }= 1 (1/4) = 3/4. a_{1000 }= 1 (1/1000) = 999/1000 is already much closer to 1. Now you can ask yourself what happens when you look at n larger and larger. Since the sequence increases monotonically, one gets as close as desired to 1 with ever larger n, but never reaches this, since 1 / n would have to become 0 for this.

Here the sequence a_{n}= 1 (1 / n) is no longer shown in the Cartesian coordinate system, but only its individual terms on the number line. To the (presumed) __limit__ a stripe is placed at a distance epsilon (a very small positive number) and the elements in the sequence that are not in it are counted. In every little one __Epsilon strips__ almost all of the elements of the sequence must be found. (almost all = all but a finite number). This is the definition of the limit.

In our example above, the limit value would be written as follows:

If a sequence of numbers has a limit, it is called __convergent__ (converging), the sequence converges towards the limit value.

If it has no limit, it is called __divergent__ (diverging), the sequence diverges. (see example 3 above)

With this so-called **Square plant** the area that is added by the small squares becomes smaller and smaller; so where does the area of the outer squares strive?

Here is the calculation with which one comes to the conclusion from above (see picture): (a is the edge length of the original square)

The area of a square of the kth generation is (a / 3^{k})^{2}.

Of a certain generation k there are 3^{k} Squares, so that the area of all squares of a generation a^{2}/3^{k} is.

For the area of all squares one must have the __total __form, and let n go towards infinity:

Consequences appear in many mathematical processes. An example are __Approximation method__. The Greek mathematician Archimedes (around 287 to 212 BC) tried to determine the relationship between the diameter and the circumference of a circle by means of complex approximations. He calculated the circumference of n-vertices, which he inscribed and circumscribed the circle, and continued his investigations up to the value n = 96!

We can just surpass this performance. The picture below shows how a 9-corner *in* as well as another *around* a circle was laid. The associated construction with The Geometer's Sketchpad allows you to drag the blue circle (above) to turn the 9-corner into a triangle, square, ... up to a 100-corner. In addition to the circumference, the area of the circle is also specified (as is well known, the ratio between the area and the radius of each circle).

The correct mathematical description for this approximation process: We define two sequences, one each for the circumference of the inner and the outer n-gon. (Because we're interested in, we cut each circumference in half.)

- Inside
_{n}= [half the circumference of the inscribed n-gon] - Outside
_{n}= [half the circumference of the circumscribed n-gon]

In addition, it is easy to see that the two sequences "nest" the limit value with increasingly good approximation, that is:

lim Inside _{n}=lim Outside _{n}=

Insidefor all n._{n}n + 1 < n + 1 n

There is little point in proving this mathematically. Conversely, however, we could try to use the convergence of the sequences to calculate approximate values for.

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