# Infinite summation SL Portfolio type I. Concerning the portfolio, the evaluation is composed on the sum of the series

Infinite summation – SL Portfolio type I

Name: Abdelrahman Lotfy Mosalam

Presented to: Mr. Hamed Mokhtar

Date: 11/10/2011

Class: DP2

The infinite series is almost considered as the main tool in calculus, it has different utilizes. It guesses the behavior of functions, investigates differential equations and also it's used in numerical analysis. Beside these uses in math, the infinite series may be used in physics and economics as well.

Concerning the portfolio, the evaluation is composed on the sum of the series  below, where:

t0 = 1, t1 = , t2 = , t3 =, … , tn =....

for the value  in the series above, the task gave a notation for the factorial , described as :

(Given)

To find out the general statement that represents the infinite sum of this sequence,  it's always required to determine the sum  of the first  terms of the infinite sequence for , for where

Using the notations given mentioned and the sequence values, we concluded this table containing the variation of  and it's effect on the

Note for ALL tables and graphs:

The only controlled variable in the table is the

All of the answers below will be corrected to 6 decimal places

Domain of n is

Graphs are all done using Microsoft Excel

Relation between  and                  (

I can notice right now from this plot, that the  value is increasing when the  value increases, so simply they are positively correlated.

however, the graph does not go beyond . So now  can completely agree that there is a kind of asymptote at moreover, it is noticeable that as , the values of . Even though on the table, the data shows that when n=10, .  In this case we can call what is happening a limitation, because now in the table we see the , while on the graph the doesn’t intersect the asymptote, this has occurred because we can only take six decimal places. So when we expended the value, It shows when

The investigation will be clearer and more will result more relevant if we change and add a new variable on the investigation. This time the variable
will be shifted into 3, to see a clearer impact on the  and have a more developed observation…

Relation between  and                 (

Again, let’s plot a graph representing the relation between  and .

This plot insure the proof that the  value is increasing while the   is also increasing. This time, the graph would not go further. It's noticeable that the asymptote ...