In this portfolio, I will determine the general sequence tn with different values of variables to find the formula to count the sum of the infinite sequence.

, the value of n increases as well, but it does not exceed 3, thus the greatest value of this relationship will be 3 and therefore the domain of this relationship is

, as n approaches

.

From the above calculations can be concluded that the greatest value of these sequences is a, however in order to support and check this I will consider general sequence x=1 and, again, calculate

 for the first ten terms (

3. ). However this time I am going to use different values of such as a=5 and a=7. Moreover I will try to find a general statement which fits this sequence.

Let

 therefore the sum of

will look the following way:

After I calculated the sums for

 I am going to represent the relation between

 on a graph:

From this graph we can see that as

, the value of n increases as well, but it does not exceed 5, thus the greatest value of this relationship will be 5 and therefore the domain of this relationship is

, as n approaches

.

Let

 therefore the sum of

will look the following way:

After I calculated the sums for

 I am going to represent the relation between

 on a graph:

From this graph we can see that as

, the value of n increases as well, but it does not exceed 7, thus the greatest value of this relationship will be 7 and therefore the domain of this relationship is

, as n approaches

.

From all of the calculations which I performed now I can find a general statement for finding the sequence of this sum (

). In order to continue with finding the general formula, first I have to simplify the original sum of infinite sequences:

. In order to do this I will replace (xlna) with m, therefore this sequence will look in the following:

Hence the sum of these infinite terms will be:

However on power series expansion’s side my general statement will have the following form:

From here follows that the general statement for the infinite sequences is

I concluded that this is the general statement, on the basis of the following facts: as

 increases the value of n will increase also

 Now I want to expand my investigation in order to find a general statement which represents the infinite sum of the infinite sequence

, where

In order to find the general statement of these infinite sequences I will define

 as the sum of the first n terms for various values of a and x.

First I will find

, where I have a=2 and I calculate x for 1,2,3,4,5,6,7,8,9,10

Therefore I will calculate the sum for the first nine terms:

In order to comment on the relation between

2x and x I need to plot on a graph:

Looking at the graph I can conclude that as x increases, the value of

 increases as well.

Now I am going to perform the same calculations as above, but with values of a=3 and x=1,2,3,4,5,6,7,8,9,10

Therefore I will calculate the sum for the first nine terms:

In order to comment on the relation between

(3,x) and x I need to plot on a graph:

Looking at the graph I can conclude that as x increases, the value of

 increases as well.

This time I want my a=4 and my x=1,2,3 and 4, and I want to have in the range of

, from calculating this sequence I will be able to find a general statement for this sequence.

In order to comment on the relation between

(4,x) and x I need to plot on a graph:

Looking at the graph I can conclude that as x increases, the value of

 increases as well.

In order to find a general statement I need to ask myself a question: How does

increase as n approaches

?

First I will test the general statement which I found above, using different values of a and x. The formula which I found earlier is

 and the values of x are 1,2,3,4 and the values of a is 4.

Therefore I will have the following results:

x=1, a=4 =>

 = 4

x=2, a=4=>

 = 16

x=3, a=4=>

 = 64

x=4, a=4=>

 = 256

From these results I can see that as x increases the value of

 increases as well, which suggest that as the value of

, the values of a and x increases too. From here I can conclude that the sum of

 have a makes up the following general statement:

.

4. In order to prove that my general statement can fit any values of a and x, I need the test validity of the general statement. I will do it through considering different values of a and x. Therefore my a will be 3,5,2,6,4 and my x will be 3,2,4,8,6

Thus using this formula I will get the following results:

a=3,x=3=>

a=3,x=2=>

a=3,x=4=>

a=3,x=8=>

a=3,x=6=>

a=5, x=3=>

a=5, x=2=>

a=5, x=4=>

a=5, x=8=>

a=5, x=6=>

a=2,x=3=>

a=2,x=2=>

a=2,x=4=>

a=2,x=8=>

a=2,x=6=>

a=6,x=3=>

a=6,x=2=>

a=6,x=4=>

a=6,x=8=>

a=6,x=6=>

a=4,x=3=>

a=4,x=2=>

a=4,x=4=>

a=4,x=8=>

a=4,x=6=>

Now I am going to plot my results on a graph and comment on the situation in the graph:

From the table which I made I can clearly see that as value of a and x increases the value of

 increases also, which support my general statement and proves the statement that as the value of

, the values of a and x increases too. Therefore my general statement can fit any values of a and x.

• We know the range of values Tn (a,x).
• We know how Tn (a,x) changes when x changes.
• We know the domain for a and x: positive numbers.
• It’s easy to find the infinite sum, just by setting values for a and x.
• We know this is power series expansion.

What I did to find out this statement is just calculating, and while doing this, I have been realising, step by step, some signs that suggest about the Tn (a,x), like range, sign and simplest formula.

After doing this portfolio, I learned several things, such as

• using mathematical technology on computer, which I did not know before;
• constructing the parts of the work so that it looks logically;
• using appropriate language when doing mathematical big work;
• realising subtleties from graph/plot rather than from statistics as before I was used to;
• how to find the sum of infinite general sequence.