The Summation Notation

4 Jan 1999

The Notation
Some Results
Examples

he Notation

In mathematics, we often use symbols to simplify mathmatical expression.

Expression like 1² + 2² + 3² + ¼ + n² is often written in short as år², which means "the sum of all terms like r² ".

To be more precise, numbers are placed above and below å to show where the series begins and ends.

Thus

1² + 2² + 3² + ¼ + n² =

n
å
r = 1

r².

In general, a finite series a_m + a_m_{+ 1} + ¼ + a_n, where m £ n, is written as

n
å
r = m

a_r .

In this notation,

r is called the index of summation (may be replaced by other symbols),
a_r is the general term of the series,
m is the lower limit,
n is the upper limit,
å is a greek capital letter (read as sigma) which corresponds to S (for sum).

For infinite series a_m + a_m_+
1 + ¼, we replace the upper limit by ¥.

So the representation is

¥
å
r = m

a_r .

Usually, the lower limit is 1, but it should not be assumed to be always the case.

n
å
r = 1

a_r

= a₁ + a₂ + ¼ + a_n .

ome Results

n
å	a_r
r = m

n
å	a_r
r = 1

m - 1
å	a_r
r = 1

where n > m ³ 1

n
å	(aU_r + bV_r)
r = m

= a

n
å	U_r
r = m

+ b

n
å	V_r
r = m

where a, b are constants

n
å	k
r = 1

= nk where k is a constant

n
å	r
r = 1

= ½n(n + 1)

n
å	r²
r = 1

= n(n + 1)(2n + 1)/6

n
å	r³
r = 1

= [n(n + 1)/2]²

n
å	k^r
r = 1

k(kⁿ - 1)

¾¾¾¾

k - 1

where k ¹ 1

xamples

Example 1: Find

n

å (r + 1)(r + 2).

r = 1

Solution:

n
å	(r + 1)(r + 2)
r = 1

n
å	(r² + 3r +2)
r = 1

n
å	r² + 3
r = 1

n
å	r +
r = 1

n
å	2
r = 1

(n/6)(n + 1)(2n + 1) + 3(n/2)(n + 1) + 2n

(n/3)[n² + 6n + 11].

Example 2: Find

2n

å (2^r - 3r²).

r = n + 1

Solution:

2n
å	(2^r - 3r²)
r = n + 1

(2^{n + 1} + 2^{n + 2} + ¼ + 2²ⁿ) - 3[

2n
å	r² -
r = 1

n
å	r²
r = 1

]

2^{n + 1}(2ⁿ - 1)
¾¾¾¾¾¾	- 3[(2n/6)(2n + 1)(4n + 1) - (n/6)(n + 1)(2n + 1)]
2 - 1

2^{2n + 1} - 2^{n
+ 1} - (n/2)(2n + 1)(7n + 1).

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