# SL Type I Mathematics Portfolio

Infinite Summation

SL Type I Mathematics Portfolio

Infinite series are among the most powerful and useful tools that you will encounter in calculus. They are among the major tools used in analyzing differential equations, developing methods of numerical analysis, defining new functions, and estimating behaviour of functions, and much more. The use of infinite series can be found in a variety of fields, such as electronics engineering, micro-economics, mathematics, and physics.

In this Mathematical portfolio, the investigation of the sum of infinite sequence, ,  where:       t0 = 1, t1 = , t2 = , t3 =, … , tn =.

For a positive integer, factorial , written as , is the product of all of the positive integers less than or equal to .

.

Furthermore,  (by definition)

This can be proofed with the equation

The above sequence is called an infinite sequence, because the three dots at the end of the sequence indicate that the sequence continues indefinitely. However, the following sequence   is called a finite sequence because it has a finite number of terms, . After determine this sequence as an infinite sequence, we should determine whether or not this sequence is convergent or divergent. This is important because if this sequence diverges, the general statement would be  .  A convergent series is a series in which the terms decrease in magnitude rapidly and for which the sum of the first several terms is not too different from the sum of all of the terms of the series. The following is an example of a divergent series.

=

Thus, there is no general statement to represent the infinite sum of this sequence. In order to detect whether the given sequence is a convergent sequence or not, there are 3 tests that can prove it. The ratio test is the most comprehensive, and useful test.

If

1. Is less than 1, the sequence converges.
2. Is greater than 1, the sequence diverges
3. Is equal to 1, the test fails to give conclusive information

We can use the ratio test to determine whether or not our sequence is converging.

Since the 0 is less than 1, the series converges. This determines that the sum of this series is a real number.

To determine the general statement that represents the infinite sum of this general sequence, we must start step by step:  (all of the answers below will be corrected to 6 decimal places)

First, we have to calculate the sum  of the first  terms of the sequence for , for where  :

Relation between  and                  (

By using Microsoft Excel, it is possible to graph the relation between  and .

I noticed that from this plot, the  value is increasing when the  value increases, however, the graph does not go beyond . We can say that there is an asymptote at Furthermore, it is noticeable that as , the values of  , which in this case is the . Even though on the table, the data shows that ...