Everything Maths Grade 12

3.9 CHAPTER 3. SEQUENCES AND SERIES

3.9 Infinite Series EMCAB

Thus far we have beenworking only with finite sums, meaning that whenever we determinedthe

sum of a series, we only considered the sum of the first n terms. In this section,we consider what

happens when we addinfinitely many terms together. You might thinkthat this is a silly question -

surely the answer will be∞ when one sums infinitely many numbers, no matter how small they are?

The surprising answer isthat while in some cases one will reach∞ (like when you try to add all the

positive integers together), there are some casesin which one will get afinite answer. If you don’t

believe this, try doing the following sum, a geometric series, on your calculator or computer:

1

2 +

1

4 +

1

8 +

1

16 +

1

32 +...

You might think that if you keep adding more and more terms you will eventually get larger andlarger

numbers, but in fact youwon’t even get past 1 - try it and see for yourself!

We denote the sum of an infinite number of terms of a sequence by

S∞=

�∞

i=1

ai

When we sum the termsof a series, and the answer we get after each summation gets closer and closer

to some number, we say that the series converges. If a series does not converge, then we say that it

diverges.

Infinite Geometric Series EMCAC

There is a simple test for knowing instantly which geometric series converges and which diverges.

When r, the common ratio, isstrictly between− 1 and 1 , i.e.− 1 < r < 1 , the infinite series will

converge, otherwise it will diverge. There is also a formula for workingout the value to whichthe

series converges.

Let’s start off with Formula (3.30) for the finite geometric series:

Sn=

�n

i=1

a 1 .ri−^1 =

a 1 (rn− 1)

r− 1

Now we will investigatethe behaviour of rnfor− 1 < r < 1 as n becomes larger.

Take r =^12 :

n = 1 : rn= r^1 = (^12 )^1 =^12

n = 2 : rn= r^2 = (^12 )^2 =^12.^12 =^14 <^12

n = 3 : rn= r^3 = (^12 )^3 =^12.^12.^12 =^18 <^14

Since r is in the range− 1 < r < 1 , we see that rngets closer to 0 as n gets larger.