Sequences and Series IV

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What is a Series?
A series is the sum of the terms of a sequence.
So for the finite sequence: 2, 5, 8, 11 the corresponding series is: 2 + 5 + 8 + 11.
In general for the Sequence:

The corresponding series is:

1. The Sigma Notation.
Instead of writing out a series in list form a shorthand notation has been devised called the Sigma Notation.
List:

Sigma:

Example 27

Write out list in the form of a series

Solution

Notes:
(i)

is the sigma sign.

(ii) There is nothing special about the r sign. Any letter can be used.

(iii) The object beside the

sign is always the r^th term, u_r, if the letter r is used.
So in

(iv) The infinite series
This is simply

You do S_n and then evaluate

2. Tricks with the sigma notation.
(i) The split trick: If u_r is the sum of functions then you can split the sum into individual functions.

(ii) The Factor Trick: You can take out constants as factors through the sigma sign.

(iii) The One Trick

3. Summation Techniques.
There are two techniques for adding up series
[A] Memory series
[B] Method of Differences

[A] Memory Series
You are expected to know the following results:
(a)

(b)

(c)

This is a special case of (a) i.e. it is an Arithmetic Series with a = 1 and d = 1.

(d)

The proof is by induction.

Note: The Geometric Series is the hardest to recognise. Look for a power in u_r and then write out the series.

Example 28

Evaluate

Solution

Notes
(i) You may be asked to do combinations of the memory series.

(ii) If you are asked to find S₄₂ find S_n and plonk in 42 at the end.

[B] Method of differences (MOD)
This technique enables us to add up the terms of other series including: Fractions, Surds, Logs, Factorials.
Steps:
(i) Write the general term as a difference of two objects.
(ii) List the terms of the series in a vertical table.
(iii) Add up the series by cancelling objects in successive rows.