Infinite Summation

–4–

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