MATME/PF/M11/N11/M12/N12

INFINITE SUMMATION

SL TYPE I

Aim: In this task, you will investigate the sum of infinite sequences tn , where t0 = 1, t1 =

( x ln a )

( x ln a ) 2

( x ln a )3

( x ln a) n

… , tn =

….

, t2 =

, t3 =

n!

1

2 ×1

3 × 2 ×1

A notation that you may find helpful in this task is the factorial notation n ! , defined by

n= n(n − 1)(n − 2)....3 × 2 × 1

!

e.g. 5! = 5 × 4 × 3 × 2 ×1 (= 120)

Note that 0 ! = 1

Consider the following sequence of terms where x = 1 and a = 2 . (ln 2) (ln 2) 2 (ln 2)3

1,

,

,

…

1

2 ×1 3 × 2 ×1

Calculate the sum S n of the first n terms of the above sequence for 0 ≤ n ≤ 10 . Give your answers correct to six decimal places.

Using technology, plot the relation between S n and n. Describe what you notice from your plot. What does this suggest about the value of S n as n approaches ∞ ? Consider another sequence of terms where x = 1 and a = 3 .

1,

(ln 3) (ln 3) 2 (ln 3)3

,

,

…

1

2 ×1 3 × 2 ×1

Calculate the sum S n of the first n terms of this new sequence for 0 ≤ n ≤ 10 . Give your answers correct to six decimal places.

Using technology, plot the relation between S n and n. Describe what you notice from your plot. What does this suggest about the value of S n as n approaches ∞ ? Now consider a general sequence where x = 1 .

1,

(ln a ) (ln a ) 2 (ln a )3

,

,

…

1

2 ×1 3 × 2 ×1

Calculate the sum S n of the first n terms of this general sequence for 0 ≤ n ≤ 10 for different values of a. Give your answers correct to six decimal places.

Using technology, plot the relation between S n and n. Describe what you notice from your plot. What does this suggest about the value of S n as n approaches ∞ ? Use your observations from these investigations to find a general statement that represents the infinite sum of this general sequence.

(This task continues on the following page)

For final assessment in 2011 and 2012

–5–

MATME/PF/M11/N11/M12/N12

Now we will expand our investigation to determine the sum of the infinite sequence tn, where ( x ln a )

( x ln a ) 2

( x ln a )3

= 1, t1

t0 =

, t2

=

=

, t3

….

1

2 ×1

3 × 2 ×1

Define Tn (a , x) as the sum of the first n terms, for various values of a and x, e.g. T9 (2, 5) is the sum of the first nine terms when a = 2 and x = 5 . Let a = 2 . Calculate T9 (2, x) for various positive values of x. Using technology, plot the relation between T9 (2, x) and x. Describe what you notice from your plot. Let a = 3 . Calculate T9 (3, x) for various positive values of x. Using technology, plot the relation between T9 (3, x) and x. Describe what you notice from your plot. Continue with this analysis to find the general statement for Tn (a , x) as n approaches ∞ . Test the validity of the general statement with other values of a and x. Discuss the scope and/or limitations of the general statement. Explain how you arrived at the general statement.

For final assessment in 2011 and 2012

Internal assessment criteria and additional notes

Additional notes on applying the criteria

These additional notes on applying the criteria were written by the senior moderators following the May 2006 examination session. They will be helpful to teachers and were included in the subject report.

Criterion A: use of notation and terminology

Achievement

level

Descriptor

0

The student does not use appropriate notation and terminology.

1

The student uses some appropriate notation and/or terminology.

2

The student uses appropriate notation and terminology in a consistent manner and does so throughout the work.

Tasks will probably be set before students are aware of the notation and/or terminology to be used. Therefore the key idea behind this criterion is to assess how well the students’ use of terminology describes the context. Teachers should provide an appropriate level of background knowledge in the form of notes given to students at the time the task is set....