ALGEBRA II H/GALGEBRA II H/G@@

INTRODUCTION TO SERIESINTRODUCTION TO SERIES

SERIES : The sum of part (partial) or all of the terms of a sequence.

The sequence 3, 8, 13, 18 can be written as the series 3 + 8 + 13 + 18.∑ is the Greek letter sigma which means summation.

5

k=12k = 21 + 22 + 23 + 24 + 25 is called

sigma notation.

1) What does k5

5

k=1S= 3 mean?

SOLUTION :

k5

5

k=1S= 3 =31 + 32 + 33 + 34 + 35

2) What does nk

n

k=1S= t mean?

SOLUTION :

nk

n

k=1S= t = t1 + t2 + t3 + t4 + … + tn

3) Evaluate.

k4

4

k=1S= 2 3

SOLUTION :

k4

4

k=1S= 2 3 = (2 • 31) + (2 • 32) +

(2 • 33) + (2 • 34)

= 6 + 18 + 54 + 162= 240

4) Evaluate.

k5

k=0-1 3k - 2

SOLUTION :

k5

k=0-1 3k - 2

= (-1)0[3(0) – 2] + (-1)1[3(1) – 2] +

(-1)2[3(2) – 2] + (-1)3[3(3) – 2] +

(-1)4[3(4) – 2] + (-1)5[3(5) – 2] = -2 + -1 + 4 + -7 + 10 + -13= -9

Write using sigma notation.

5) S30 = 1 + 8 + 27 + 64 + 125 + …

SOLUTION : Hey, those numbers are all cubes.

330

k=1k

6) S50 = 3 + 6 + 12+ 24 + 48 + …

SOLUTION : Hmmm, the numbers are all multiple of 3, so let’s factor out a 3.3(1 + 2 + 4 + 8 + 16 + …) Holy cow,

look at the numbers in the (), they are all powers of 2.

n(3 2 )49

n=0OR n 1(3 2 )

50

n=1

WEB LINKShttp://www.math.montana.edu/frankw/ccp/general/sigma/learn.htm

http://quiz.econ.usyd.edu.au/mathquiz/sigma/index.php