Page 9 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

lim 1 3n + 1 2 = lim 1 3 + 1 2
n → ∞ n n → ∞ n
1
lim
lim
= n → ∞ 3 + n → ∞ 1 2
n
= 3 + 0
= 3

The summation notation ∑

Suppose we want to find the sum of the series of numbers
2 1 + 2 + 3 + … + 100

One simpler way of representing the sum of a series such as this is by using the Greek alphabet ∑, read as
100
“sigma”. For example, we write ∑ r to represent the sum 1 + 2 + 3 + … + 100, where the r term, u , is r, i.e.
th
r = 1 r
100 100
∑ u = ∑ r = 1 + 2 + 3 + … + 100.
r = 1 r r = 1
200
Similarly, we write ∑ to represent the sum 101 + 102 + 103 + … + 200, i.e.
r = 101
200
∑ r = 101 + 102 + 103 + … + 200.
r = 101
Notice that
101 + 102 + … + 200 = (100 + 1) + (100 + 2) + … + (100 + 100)
So, we can also write
200 100
∑ r = ∑ (100 + r).
r = 101 r = 1
In general, if u is the r term of a series, then
th
r
n
∑ u = u + u + u + … + u
r = 1 r 1 2 3 n
Notice that if k is a constant,
n
∑ (ku ) = ku + ku + … + ku n
r
1
2
r = 1 = k(u + u + … + u )
n
2
1
n
= k ∑ u
r = 1 r
When u = u = u = … = u = k, then
2
3
1
n
n
∑ k = k + k + … + k
r = 1 = nk
n
∑ (u + v ) = (u + v ) + (u + v ) + … + (u + v )
r = 1 r r 1 1 2 2 n n
= (u + u + … + u ) + (v + v + … + v )
n
n
1
1
2
2
n n
= ∑ u + ∑ v
r = 1 r r = 1 r
n n n
Similarly, ∑ (u – v ) = ∑ u – ∑ v
r = 1 r r r = 1 r r = 1 r
96
02 STPM Math T T1.indd 96 3/28/18 4:21 PM