Infinite Summation- The Aim of this task is to investigate the sum of infinite sequences tn.

by cfish (student)

Infinite Summation

Math Portfolio

Introduction:

Aim of this task is to investigate the sum of infinite sequences tn. An infinite sequence is a listing of numbers with no limits, like:

(1;2;3;4;5;…)

Whereas the three dots (…) at the end of the sequence states the infinity. The sum of an infinite sequence can be determined, depending on the sequence. Sequences are categorised in geometric sequences and arithmetic sequences, where in both notation and calculations differ from each other. An arithmetic sequence is a series of numbers, where the common difference (d) between terms is constant; the common difference is added or subtracted to the term before. A geometric sequence on the other hand, has a common ratio (r), where r will be multiplied to the term before, which also is constant. For both types of sequences, it is possible to calculate specific terms and the sum to certain point, whereas the number of terms is given by (n).

In this task the infinite sequence of tn, will be examined, where

t0=1, t1=(xlna)1,t2=(xlna)22×1,t3=(xlna)33×2×1 …,tn=(xlna)nn!…

As the general sequence states n!, which is factorial notation and is used to simplify. Factorial notation (n!), is defined by

n!=nn-1n-2…3×2×1          Note that 0!=1

To observe, how the sum of the sequence changes, it will be assumed that x=1 and a=2,

1, (ln2)1,(ln2)22×1,(ln2)33×2×1…

The sum of the first 11 terms, 0≤n≤10, starting with S0 and ending with S10, will be calculated with the help of the GDC. The t th term will be used to indicate the used term and Sn states the sum of the term given by n. (all in 6 decimal places, whereas the space after the 6 decimal indicates the minor differences between the last 4 sums).

S0= t0=1

S1= 1+t1= 1.693147

S2=S1+t2=1.933374

S3= S2+t3=1.988878

S4= S3+t4=1.998496

S5= S4+t5=1.999829

S6= S5+t6=1.999983

S7= S6+t7=1.999998 569

S8= S7+t8=1.999999 891

S9= S8+t9=1.999999 992

S10= S9+t10=1.999999 999 5

This diagram shows the relation between Sn and n using the gained results. (Diagram Microsoft Excel)

The diagram clearly shows, that when n increases, the value of Sn increases as well. The first 4 n values show a steady increase, but as n approaches 5 the value of Sn only increase minimally, which suggests that S ,when x=1 and a=2, will be equal to 2. This shows a relation between a and ∞, when a equals 2 the maximum for Sn (S) will also equal to 2. So this would mean that the domain of the function is 1≤Sn≤2.

Another sequence of terms will be used to underline the observations from the sequence before, this will be done the exact same way as before. For the second sequence of terms, x=1 and a=3.

1, (ln3)1,(ln3)22×1,(ln3)33×2×1…

Again, the sum of the first 11 terms, 0≤n≤10, starting with S0 and ending with S10, will be calculated with the help of the GDC. The same method and notation will be used, as in the sequence before.

S0= t0=1

S1= 1+t1=2.098612

S2=S1+t2=2.702087

S3= S2+t3=2.923081

S4= S3+t4=2.983779

S5= S4+t5=2.997115

S6= S5+t6=2.999557

S7= S6+t7=2.999940

S8= S7+t8=2.999993

S9= S8+t9=2.999999 217

S10= S9+t10=2.999999 923

This diagram shows the relation between Sn and n using the gained results. (Diagram Microsoft Excel)

Now again, it is possible to see relating on the diagram and the table, that when n increases, the value of Sn increases as well. This time the first 5 values of n show a visible steady increase. The complete data collection for the equation, suggests that when n is ∞, when x=1 and a =3, the sum will equal to 3. This may suggest that the ...