Again, I noticed that when and , the values of Sn increase as values of n increase, but don’t exceed 3. So it suggests that Sn will be in domain 1 Sn 3 as n approaches .
Above, I have been supposing that the greatest value for the sum of infinite sequence Sn is a. And I want to check if it’s correct with some different values of a.
Considering the general sequence where , I will calculate the sum Sn of the first (n+1) terms for for different values of a.
So, I will take random values for a, for example, and . And I will do exactly as above with and to see if there is any general statement for Sn .
Consider the sequence where and :
Using GDC, I will calculate the sums S0, S1, S2, …, S10:
S0 = t0 = 1
S1 = S0 + t1 = 1 + = 2.945910
S2 = S1 + t2 = 2.945910 + = 4.839193
S3 = S2 + t3 = 4.839193 + = 6.067246
S4 = S3 + t4 = 6.067246 + = 6.664666
S5 = S4 + t5 = 6.664666 + = 6.897171
S6 = S5 + t6 = 6.897171 + = 6.972577
S7 = S6 + t7 = 6.972577 + = 6.993539
S8 = S7 + t8 = 6.993539 + = 6.998638
S9 = S8 + t9 = 6.998638 + = 6.999740
S10 = S9 + t10 = 6.999740 + = 6.999956
Using Microsoft Excel, I plot the relation between Sn and n:
And I noticed that Sn increases when n increases, and values of Sn seem like not to exceed 7. So, the domain for infinite sum of the general sequence where and is suggested to be 1 Sn 7.
Now, moving on to the second example, I will consider the sequence where and :
So using GDC, I calculate the sums S0, S1, S2, …, S10:
S0 = t0 = 1
S1 = S0 + t1 = 1+ = 2.144730
S2 = S1 + t2 = 2.144730 + = 2.799933
S3 = S2 + t3 = 2.799933 + = 3.049943
S4 = S3 + t4 = 3.049943 + = 3.121492
S5 = S4 + t5 = 3.121492 + = 3.137873
S6 = S5 + t6 = 3.137873 + = 3.140998
S7 = S6 + t7 = 3.140998 + = 3.141509
S8 = S7 + t8 = 3.141509 + = 3.141582
S9 = S8 + t9 = 3.141582 + = 3.141591
S10 = S9 + t10 = 3.131591 + = 3.141592
Then, using Microsoft Excel, I plot the relation between Sn and n:
Knowing that (correct to six decimal places), I noticed that in the sequence given where and , Sn increases as n increases, and doesn’t exceed . So domain for the infinite sum Sn here is again suggested to be 1 Sn .
Now let’s analyse the initial general sequence:
t0 = 1, t1 = , t2 = , t3 = , … , tn = , ….
If I substitute (xlna) with m, I can have a sequence like this:
t0 = 1, t1 = m, t2 = , t3 = , …
And the sum of these infinite terms is:
On the other hand, as defined by power series expansion, we have:
Therefore, we see that the infinite sum can be counted by em, where m = xlna.
So what I noticed from here is (hypothesis):
-
Values of Sn increase as values of n increase.
-
The greatest value for Sn infinite is suggested to be a. 1 Sn a.
-
The statement to find the sum of infinite sequence is suggested to be e(xlna).
Now, it would be very interesting to expand this investigation to determine the sum of the infinite sequence tn, where:
t0 = 1, t1 = , t2 = , t3 = , … , tn = , …
Tn (a,x) is defined to be the sum of the first n terms, for variable values of a and x.
E.g.: T6 (2,3) is the sum of the first 6 terms when a = 2 and x = 3.
Let a = 2. I will calculate T9 (2,x) for various positive values of x, for example: 1,2, 3, 4, 5, 6, 7, 9,10,11.
Firstly, using GDC, I will calculate the sum of the first 9 terms when a = 2 and x = 1:
T9 (2,1) = t0 + t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8
=
= 1.999998322
When a = 2 and x = 2:
T9 (2,2) =
= 3.999983
When a = 2 and x = 3:
T9 (2,3) =
= 7.997486
When a = 2 and x = 4:
T9 (2,4) =
= 15.963512
When a = 2 and x = 5:
T9 (2,5) =
= 31.702131
When a = 2 and x = 6:
T9 (2,6) =
= 62.305296
When a = 2 and x = 7:
T9 (2,7) =
= 120.465723
When a = 2 and x = 9:
T9 (2,9) =
= 420.699406
When a = 2 and x = 10:
T9 (2,10) =
= 755.692615
When a = 2 and x = 11:
T9 (2,11) =
= 1320.526575
Using Microsoft Excel, I will plot the relation between T9 (2,x) and x:
From this plot, I can observe that as x increases, T9 (2,x) is suggested to increase.
Now, let a = 3. I will calculate the sum T9 (3,x) for various positive values of x, for example: 1, 2, 3, 4, 5, 6, 7, 8, 9, 12.
Using GDC, I am going to calculate the sum of the first 9 terms when a = 3 and x = 1:
T9 (3,1) = t0 + t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8
=
= 2.999991
When a = 3 and x = 2:
T9 (3,2) =
= 8.995813
When a = 3 and x = 3:
T9 (3,3) =
= 26.814822
When a = 3 and x = 4:
T9 (3,4) =
= 78.119155
When a = 3 and x = 5:
T9 (3,5) =
= 217.471547
When a = 3 and x = 6:
T9 (3,6) =
= 569.033838
When a = 3 and x = 7:
T9 (3,7) =
= 1390.256866
When a = 3 and x = 8:
T9 (3,8) =
= 3174.042570
When a = 3 and x = 9:
T9 (3,9) =
= 6802.981104
When a = 3 and x = 12:
T9 (3,12) =
= 48714.21475
Now, using Microsoft Excel, I plot the relation between T9 (3,x) and x:
Here, I also see that values of T9 (3,x) increase as values of x increase, T9(3,x) seems to increase.
To find the general statement for Tn (a,x) as n approaches , I will continue with this analysis until realising appropriate formula.
So let’s take another value for a and see if we get the same notation as above.
For example, let a = , using GDC, I will calculate T7 for variable positive values of x. Let’s take 1, 2, 3, 4, 5, 6, 7, 9, 11, 13:
When a = and x = 1:
T7 (,1) = t0 + t1 + t2 + t3 + t4 + t5 + t6
=
= 3.140998
When a = and x = 2:
T7 (,2) =
= 11.579476
When a = and x = 3:
T7 (,3) =
= 29.135563
When a = and x = 4:
T7 (,4) =
= 79.954196
When a = and x = 5:
T7 (,5) =
= 199.094837
When a = and x = 6:
T7 (,6) =
= 451.373522
When a = and x = 7:
T7 (,7) =
= 941.654242
When a = and x = 9:
T7 (,9) =
= 3344.217931
When a = and x = 11:
T7 (,11) =
= 9647.896312
When a = and x = 13:
T7 (,13) =
= 23886.45538
And now, using Microsoft Excel, I will plot the relation between T7 (,x) and x:
From the plot, I observed that T7 (,x) increases as x increases.
What I want to find out is: How does Tn (a,x) increase as n approaches ?
So let’s test the validity of the statement found above (exlna) by counting the infinite sum for variable a and x.
When x = 1 and a = 2: Sn = eln2 = 2 = 21
When x = 2 and a = 2: Sn = e2ln2 = 4 = 22
When x = 3 and a = 2: Sn = e3ln2 = 8 = 23
When x = 4 and a = 2: Sn = e4ln2 = 16 = 24
When x = 1 and a = 5: Sn = eln5 = 5 = 51
When x = 2 and a = 5: Sn = e2ln5 = 25 = 52
When x = 3 and a = 5: Sn = e3ln5 = 125 = 53
When x = 4 and a = 5: Sn = e4ln5 = 625 = 54
➔ That is so interesting, that I noticed :
-
Tn (a,x) increases when a or/and x increases.
-
The sum of terms of this infinite sequence equals exlna, which then I realised is equal ax.
To test more about this with other values of a and x, I need to take some other values of a and x, then calculate Tn (a,x) to check the validity of general statement written above.
So let’s take a = 1.5 and x = 8
a = 1.5 and x = 9
a = 1.5 and x = 10
a = 1.5 and x = 11
a = 1.5 and x = 12
Values of a and x are positive, x is increasing.
I will calculate 3 different sums (T3, T4 and T10) in these 5 cases (8, 9, 10, 11, 12).
When a = 1.5 and x = 8:
T3 (1.5, 8) = t0 + t1 + t2
=
= 9.504583
T4 (1.5, 8) = t0 + t1 + t2 + t3
=
= 15.192839
T10 (1.5, 8) = t0 + t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9
= +
= 25.579143
When a = 1.5 and x = 9:
T3 (1.5, 9) = t0 + t1 + t2
=
= 11.307465
T4 (1.5, 9) = t0 + t1 + t2 + t3
=
= 19.406564
T10 (1.5, 9) = t0 + t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9
= +
= 38.273650
When a = 1.5 and x = 10:
T3 (1.5, 10) = t0 + t1 + t2
=
= 13.274749
T4 (1.5, 10) = t0 + t1 + t2 + t3
=
= 24.384625
T10 (1.5, 10) = t0 + t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9
= +
= 57.152951
When a = 1.5 and x = 11:
T3 (1.5, 11) = t0 + t1 + t2
=
= 15.406434
T4 (1.5, 11) = t0 + t1 + t2 + t3
=
= 30.193679
T10 (1.5, 11) = t0 + t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9
= + +
= 85.097411
When a = 1.5 and x = 12:
T3 (1.5, 12) = t0 + t1 + t2
=
= 17.702522
T4 (1.5, 12) = t0 + t1 + t2 + t3
=
= 36.900388
T10 (1.5, 12) = t0 + t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9
= + +
= 126.215786
Now, using Microsoft Excel, I plot the relation between T3, T4, T10 and x:
So after testing what I have noticed before, here I still see that when a and x are positive, if value of x increases, Tn(a,x) also increases.
So what if x is negative? To be sure that the statement is true in all cases, there is a need to also test some negative values of x. Let’s take a = 2, and 4 different negative values of x: -8, -7, -3, -2 (ascending order). Using GDC, I will calculate T9 (2,x) and see the relation between T9 (2,x) and x.
When a = 2 and x = -8:
T9 (2, -8) = t0 + t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8
=
= 8.679707
When a = 2 and x = -7:
T9 (2, -7) =
= 2.743859
When a = 2 and x = -3:
T9 (2, -3) =
= 0.016654
When a = 2 and x = -2:
T9 (2, -2) =
= 0.250046
Using Microsoft Excel, I will plot the relation between T9 (2,x) and x (negative) :
So, here I haven’t noticed any sign of similarity to the cases before with x-positive. As x increases, Tn decreases then increases. So for the general statement found above, we have to note that it’s untrue for x-negative.
Now, let’s point out a little bit about the scope of the general statement:
-
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.
I DECLARE THAT THE WHOLE WORK IS ENTIRELY DONE ON MY OWN.
