GEOMETRIC SERIES

There is a legend that a Persian invented chess to give interest to the life of the king who was bored. For his reward, this Perian asked for a quantity of grain, according to the following rules.

1 grain was to be placed on the 1st square of the chess board, 2 on the next, 4 on the 3rd and so on, doubling the number each square. How many must be placed on the 64th square?

Geometric Sequence We have a sequence: Each term is twice the previous term, so by the 64th term we have multiplied by 2 sixty-three times We have approximately or 9 followed by 18 zeros!

The sequence is an example of a Geometric sequence A sequence is geometric if where r is a constant called the common ratio In the above sequence, r = 2

A geometric sequence or geometric progression (G.P.) is of the form The nth term of an G.P. is

Exercises 1. Use the formula for the nth term to find the term indicated of the following geometric sequences (a) Ans: (b) Ans: (c) Ans:

The formula will be proved next but you don’t need to learn the proof. Summing terms of a G.P. e.g.1 Evaluate Writing out the terms helps us to recognize the G.P. Although with a calculator we can see that the sum is 186, we need a formula that can be used for any G.P. The formula will be proved next but you don’t need to learn the proof.

Move the lower row 1 place to the right (Combining like terms) Summing terms of a G.P. With 5 terms of the general G.P., we have Multiply by r: Subtracting the expressions gives Move the lower row 1 place to the right (Combining like terms)

Summing terms of a G.P. With 5 terms of the general G.P., we have Multiply by r: Subtracting the expressions gives and subtract

Summing terms of a G.P. With 5 terms of the general G.P., we have Multiply by r: Subtracting the expressions gives

Summing terms of a G.P. So, Take out the common factors and divide by ( 1 – r ) Similarly, for n terms we get

Sum of a FINITE Geometric Series. Number of terms First term Rate “Common Ratio” Sum of “n” terms What is “missing”? There is no last term

Summing terms of a G.P. gives a negative denominator if r > 1 The formula Instead, we can use To get this version of the formula, we’ve multiplied the 1st form by

Summing terms of a G.P. For our series Using

Summing terms of a G.P. e.g. 2 Find the sum of the first 20 terms of the geometric series, leaving your answer in index form Solution: We’ll simplify this answer without using a calculator

Summing terms of a G.P. There are 20 minus signs here and 1 more outside the bracket!

Evaluate the geometric series: 4 + 2 + 1 + ½ + … Summing terms of a G.P. Evaluate the geometric series: 4 + 2 + 1 + ½ + … For the first 10 terms

Evaluate the geometric series: 4 + 2 + 1 + ½ + … Summing terms of a G.P. Evaluate the geometric series: 4 + 2 + 1 + ½ + … For the first 10 terms

Evaluate the geometric series: 4 + 2 + 1 + ½ + … Summing terms of a G.P. Evaluate the geometric series: 4 + 2 + 1 + ½ + … For the first 10 terms

Evaluate the geometric series: 4 + 2 + 1 + ½ + … Summing terms of a G.P. Evaluate the geometric series: 4 + 2 + 1 + ½ + … For the first 10 terms

Evaluate the geometric series: 4 + 2 + 1 + ½ + … Summing terms of a G.P. Evaluate the geometric series: 4 + 2 + 1 + ½ + … For the first 10 terms

Evaluate the geometric series: 4 + 2 + 1 + ½ + … Summing terms of a G.P. Evaluate the geometric series: 4 + 2 + 1 + ½ + … For the first 10 terms

Evaluate the geometric series: 4 + 2 + 1 + ½ + … Summing terms of a G.P. Evaluate the geometric series: 4 + 2 + 1 + ½ + … For the first 10 terms

Evaluate the geometric series: 4 + 2 + 1 + ½ + … Summing terms of a G.P. Evaluate the geometric series: 4 + 2 + 1 + ½ + … For the first 10 terms

SUMMARY A geometric sequence or geometric progression (G.P.) is of the form The nth term of an G.P. is The sum of n terms is or

GEOMETRIC SERIES

Suppose we have a 2 metre length of string . . . . . . which we cut in half We leave one half alone and cut the 2nd in half again . . . and again cut the last piece in half

Continuing to cut the end piece in half, we would have in total In theory, we could continue for ever, but the total length would still be 2 metres, so This is an example of an infinite series.

The series is a G.P. with the common ratio . Even though there are an infinite number of terms, this series converges to 2. Number of terms, n Sum

The series is a G.P. with the common ratio . Even though there are an infinite number of terms, this series converges to 2.

Where does each term approach? What happens to each term in the series?

This is NOT hitting the x-axis Where does each term approach? What happens to each term in the series? The series This is NOT hitting the x-axis

The series The General Term is:

An infinite series is an expression of the form:

Summing terms of a G.P.

Converging or Diverging?

Converging or Diverging?

Converging or Diverging? To where?

y= 1.5

Arithmetic and Geometric Series  

Definition of a Geometric Series        

                          Why do 1, 3, 4 and 6 Diverge?

We will find a formula for the sum of an infinite number of terms of a G.P. This is called “the sum to infinity”, e.g. For the G.P. we know that the sum of n terms is given by As n varies, the only part that changes is . This term gets smaller as n gets larger.

If |r| < 1, then the infinite geometric series The sum of the terms of an infinite geometric sequence is called a geometric series. If |r| < 1, then the infinite geometric series a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . . has the sum

Example: Find the sum of The sum of the series is

The sum of a finite geometric sequence is given by 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a1 = 5

Convergent and Divergent Series If the infinite series has a sum the series is convergent. If the series is not convergent, it is divergent.

Ways To Determine Convergence/Divergence 1. Arithmetic – since no sum exists, it diverges 2. Geometric: If |r| > 1, diverges If |r| < 1, converges since the sum exists

Example Determine whether each arithmetic or geometric series is convergent or divergent. 1/8 + 3/20 + 9/50 + 27/125 + . . . r=6/5  |r|>1  divergent 18.75+17.50+16.25+15.00+ . . . Arithmetic series  divergent 65 + 13 + 2 3/5 + 13/25 . . . r=1/5  |r|<1  convergent

Other Series When a series is neither arithmetic or geometric, it is more difficult to determine whether the series is convergent or divergent.

Convergent Geometric Series   Where a is the first term and r is the constant ratio.

As n approaches infinity, approaches zero. We write: So, for , For the series

However, if, for example r = 2, As n increases, also increases. In fact, The geometric series with diverges There is no sum to infinity

Convergence If r is any value greater than 1, the series diverges. Also, if r < -1, ( e.g. r = -2 ), So, again the series diverges. If r = 1, all the terms are the same. If r = -1, the terms have the same magnitude but they alternate in sign. e.g. 2, -2, 2, -2, . . . A Geometric Series converges only if the common ratio r lies between -1 and 1. for This can also be written as

e.g. 1. For the following geometric series, write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity. Solution: so r does satisfy -1 < r < 1 The series converges to

SUMMARY A geometric sequence or geometric progression (G.P.) is of the form The nth term of an G.P. is The sum of n terms is or The sum to infinity is or

Exercises 1. For the following geometric series, write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity. Ans: (a) so the series diverges. (b) so the series converges.