# Sigma Notation

How to write and read the given series in sigma notation: definition, 3 examples, and their solutions.

## Definition

Sigma notation is a way

to write the sum of a sequence:

a series.

[Σ] is a greek capital letter [sigma].

The given summation means (and is read as)

[the sum of a_{k} as k goes from 1 to n].

## Example4^{3} + 5^{3} + 6^{3} + ... + 19^{3}

4^{3} + 5^{3} + 6^{3} + ... + 19^{3}

This is a series.

The cubed numbers go from 4 to 19.

So write

Σ,

k = 4 at the bottom of the sigma,

and 19 at the top of the sigma.

See the terms.

4^{3}, 5^{3}, 6^{3}, ..., 19^{3}

Then a_{k} = k^{3}.

So write a^{k}: k^{3}.

Sequence

So this is the answer.

## Example1 + 3 + 5 + 7 + 9 + 11 + 13

To write the series in sigma notation,

find a_{k}.

See the terms of the series:

1, 3, 5, 7, ... .

The first term, a, is 1.

1 + 2 = 3

3 + 2 = 5

5 + 2 = 7

So write, the d, +2

between the terms.

This is an arithmetic sequence.

a = 1

d = 2

So a_{k} = 1 + (k - 1)⋅2.

Arithmetic Sequence

+(k - 1)⋅2 = +2k - 2

1 + 2k - 2 = 2k - 1

So a_{k} = 2k - 1.

See the terms of the given series.

There are 7 terms.

So k goes from 1 to 7:

(given) = a_{1} + a_{2} + ... + a_{7}.

a_{k} = 2k - 1

k goes from 1 to 7.

Then the given series is

Σ

k goes from 1 to 7

2k - 1.

So this is the answer.

## Example∑ 1/[k(k + 1)]

(k + 1) - k = 1: constant.

So

1/[k(k + 1)] = [1/[(k + 1) - k]]⋅[1/k - 1/(k + 1)].

Partial Fraction Decomposition: Formula

1/[(k + 1) - k] = 1/1

[1/1]⋅[1/k - 1/(k + 1)] = [1/k - 1/(k + 1)]

Expand the given series.

k = 1

(1/1 - 1/2)

k = 2

+(1/2 - 1/3)

k = 3

+(1/3 - 1/4)

Write ... .

k = 99

+(1/99 - 1/100)

So the given series is

(1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/99 - 1/100).

Cancel -1/2 and 1/2.

Cancel -1/3 and 1/3.

Cancel -1/4 and 1/4.

...

And cancel -1/99 and 1/99.

Then 1/1 and -1/100 remain.

So write 1/1 - 1/100.

When cancelling like this,

the remaining terms are located symmetrically.

(In this case,

the first and the last terms remain.)

1/1 = 100/100

100/100 - 1/100 = 99/100

So 99/100 is the answer.