When n=10, we can see that S n is equal to 2.000000. The graph shows us that as n tends towards ∞, S n increases and tends towards the value of 2.000000.

Now, let us look at the same equation when x=1 and a=3 as the value of n increases.

Again, as n tends towards ∞, S n increases and tends towards the value of 3.000000.

We have tried the sequence in relation to positive integers. Now, let us investigate the sequence in relation to a few other values of a as x=1. We shall try it with an irrational number, a negative number, a non-integer, 0, as well as. Firstly, let us try an irrational number.

##
Irrational number: When x=1 and a=2

Once again, we can see that as n approaches ∞, Sn approaches and reaches √2.

##
Negative number: When x=1 and a=-3

We are unable to get proper values for either tn or Sn. This is possibly because the logarithms of negative numbers do not exist. To test this theory, let us try with 0 and 1, as the value of [xln(1)]n is undefined for both numbers as well.

## 0 and 1:

When x=1 and a=0 When x=1 and a=1

As we can see, the above tables prove that the sequence is only applicable to numbers for which the value of [xln(1)]n is defined.

##
Non-integer: When x=1 and a=0.87

Once again, we see that even for non-integers, while the value of Sn does fluctuate a little, it does eventually reach 0.87 as n tends towards ∞.

From the above examples, we can conclude that a general statement can be made regarding the series, i.e. when x=1, Sn →a as n→∞.

## Proving the general statement

Now let us try to prove the general statement for the series for when x=1, using something more concrete than mere observation. To do this, let us make use of the Maclaurin series^{}, explained in detail in the scope section of the portfolio. We will be using the exponential form of the series:

ey=1+y1+y22!+y33!+…+y22!+…

Substituting the values with those of the sequence above:

exlna=1+xlna1+(xlna)22!+(xlna)33!+…+(xlna)nn!+…

When x=1:

elna=1+lna1+(lna)22!+(lna)33!+…+(lna)nn!+…

Let elna=p

ln both sides:

lnalne=lnp

∴lna=lnp

∴p=elna

=a

## Part II

To arrive at the above general statement, we varied the value of a. Now, let us try varying the value of x to see if we can arrive at a similar general statement. Let us take Tn (a,x) as the sun of the first n terms for different values of a and x. Now, let us consider the case where a=2.

##
T9 (2,x)

From the above values of the above table, we can see for each value of x, the value of T9 (2,x) approaches the value of ax. For example, when x=5, T9 (2,x) tends towards 32, that is 25. However, starting from when x=6, the value of T9 (2,x) begins to deviate significantly from the value of ax.

##
T9 (3,x)

Now let us look at T9 (3,x)

T9 (3,x)

Once again, we can observe that the graph of T9 (3,x) also resembles an exponential graph of 3x. However, again the graph begins to deviate from the graph of 3x at around when x=6.

## Validating the General Statement

To try and get values closer to those of ax, let us try increasing the number of terms.

By plotting the graphs of T9(2,x), T16(2,x) & 2x above, we can see that by increasing the number of terms used, we can get values that are much closer to the values of 2x, leading us to the general statement that as n→∞, Tn(2,x)→ax.

To reinforce the validity of the statement, let us look at two more example; when

###
a= 6

###
When a=23

## Scope

The Maclaurin series, referred to earlier, is a variation of the Taylor series. The Taylor series is “a representation of a function as an infinite sum of terms that are calculated from the values of the function's at a single point”^{} x=a:

^{}

It was introduced by English mathematician Brook Taylor in 1715. The Maclaurin series is a special case of the Taylor series, used extensively by Scottish mathematician Colin Maclaurin, for whom it was named after. Basically, if the Taylor series is centered at zero, it is considered a Maclaurin series.

The Maclaurin series is used in the Math Portfolio. The function used is the exponential function, as when Tn (a,x) is plotted, it gives us an exponential curve. Now let us mathematically derive fx=ex:

According to the Maclaurin series,

^{}

Expanding fx=ex:

Recall that the is f '(x) = ex. In fact, all the derivatives are ex.

f '(0) = e0 = 1

f ''(0) = e0 = 1

f '''(0) = e0 = 1

We see that all the derivatives, when evaluated at x = 0, give us the value 1.

Also, f(0) = 1, so we have:

The Maclaurin Series expansion will be simply:

^{}

The Maclaurin series allows us to give quick approximations of various functions, enabling the approximation of sine functions, π and other hard to calculate functions. It has many uses in the world today, probably the most common being in the programming of calculators, as they store the series expansion such that values can be substituted in to give quick calculations. However, the series does not work all the time.

## Limitations

As discovered earlier, the natural logarithms for negative numbers and 0 do not exist. The natural logarithm for 1 is also undefined as it is equal to 0. Let us now prove it mathematically:

##
For when a is a negative number:

tn=(xlna)nn!

t0=(xln(-1))00!

Let x=1

xln-1=ln-11

=0

∴t0=000!

The above value is undefined, thus the series does not work for negative numbers.

##
For when a is equal to 0:

tn=(xlna)nn!

t0=(xln0)00!

Again,

t0=000!

Which is undefined, thus the series does not work for when a is equal to 0 either.

##
For when a is equal to 1:

tn=(xlna)nn!

t0=(xln1)00!

Again,

t0=000!

Thus, we can prove that the series expansion does not work for when a=0, a=1 or

a<0.

## Conclusion

The Mathematics SL Type 1 Portfolio has helped me in many ways. As someone who is not all that great at mathematics, it has helped me tremendously to learn how much the use of technology can aid in the study of mathematics. I also learnt a great deal about infinite summation, the Taylor and Maclaurin series as well as about how they help us even in or daily life. Also, the portfolio has helped me analyze and rethink many of my approaches to mathematics thus far. Thus, from this portfolio I have not only learnt about infinite summation, but a great deal about myself and the influence mathematics has on the world around me as well.