Sigma Notation

Sigma Notation

How to solve sigma notation problems: meaning, examples, and their solutions.

Meaning

Sigma notation is a simple way to write the sum of a sequence. The given summation means 'the sum of ai as i goes from 1 to n'.

Sigma notation is a simple way
to write the sum of a sequence (= series).

'Σ' is the greek capital letter 'sigma'.

The given summation means (and is read as)
'the sum of ai as i goes from 1 to n'.

Example 1

Write the given series in sigma notation. 1/(1*2) + 1/(2*3) + 1/(3*4) + ... + 1/(99*100)

Find ak.

1/[1⋅(1 + 1)]
1/[2⋅(2 + 1)]
1/[3⋅(3 + 1)]
...
1/[99⋅(99 + 1)]

So ak = 1/[k(k + 1)].

1/[1⋅(1 + 1)]
1/[2⋅(2 + 1)]
1/[3⋅(3 + 1)]
...
1/[99⋅(99 + 1)]

So k goes from 1 to 99.

So the given series is

the sum of 1/[k(k + 1)]
as k goes from 1 to 99.

Example 2

Write the given series in sigma notation. 4^3 + 5^3 + 6^3 + 7^3 + ... + 19^3

Find ai.

43, 53, 63, ... 193

So ai = i3.

43, 53, 63, ... 193

So i goes from 4 to 19.

So the given series is

the sum of i3
as i goes from 4 to 19.

Example 3

Write the given series in sigma notation. 1 + 3 + 5 + 7 + 9 + 11 + 13

Find an.

The given series is an arithmetic series
whose a1 is 1 and d is +2.

So an = 1 + (n - 1)⋅2
= 2n - 1.

Arithmetic sequences

There are 7 terms.

So n is from 1 to 7.

So the given series is

the sum of '2n - 1'
as n goes from 1 to 7.

Example 4

Find the value of the given series. The sum of (2i - 1) as i goes from 1 to 10

an = 2n - 1

So the sigma notation shows the arithmetic series.

To use the arithmetic series formula,
find a1, d, and n.

a1 = 1
d = 2 (coefficient of n)
n = 10 (i is from 1 to 10.)

Arithmetic sequences

Use the arithmetic series formula.

(given) = S10
= (10/2)[2⋅1 + (10 - 1)⋅2]

Example 5

Find the value of the given series. The sum of i^x as x goes from 1 to 100 (i = sqrt(-1))

Find the pattern of the terms.

i1 = i
i2 = -1
i3 = -i
i4 = +1

i5 = i4⋅1 + 1 = i
i6 = i4⋅1 + 2 = -1
i7 = i4⋅1 + 3 = -i
i8 = i4⋅2 + 0 = +1

'i, -1, -i, 1' is repeating.

i100 = i4⋅25 + 0 = 1
So 'i, -1, -i, 1' is repeating 25 times.

Powers of i

i - 1 - i + 1 = 0

So each 'group' is 0.

So the sum of the 'groups' is 0.