 13 - 6   Sigma Notation

Try the quiz at the bottom of the page!
go to quiz  The Greek letter, sigma, shown above, is very often used in mathematics to represent the sum of a series.  It's a nice shorthand notation!!  For example is shorthand for the series starting with the first term and ending with the ninth term of 3k.  That is as follows:
= 3(1) + 3(2) + 3(3) + 3(4) + 3(5) + 3(6) + 3(7) + 3(8) + 3(9)
The symbol 3k is called the summand,  the numbers 1 and 9 are the limits of the summation, and the symbol k is the index.  The choice of the letter used for the index is up to you, but must match with the letter used in the summand! Properties of Infinite Series This allows you to add the sums one of two ways.  You can add the individual terms first and then sum all of them or you can sum the indivdual terms and add the two answers.  This allows you to either multiply each term by c then add the series, or first add the series and then multipy the result by c.  Your job in this section, is to learn to write sigma notation in expanded form and vice versa. Sample Problems
Write each series in expanded form.  Answers are down the page.  Don't look first!

1) 2) 3) Write each series using sigma notation!
4)  1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81
5)  5 + 9 + 13 + 17 + 21 + 25 + 29 + 33
6)  1 - 1/3 + 1/9 - 1/27 + 1/81 - 1/243  1)  The series starts with the first term and ends with the tenth term. It is summing 9k.  Replace the numbers 1 through 10 in for k.
9 + 18 + 27 + 36 + 45 + 54 + 63 + 72 + 81 + 90
(note)  If you actually had to find the sum, it would be much easier to sum the k values and then multiply by 9.
9(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) = 9(55) = 495!
2)  This series alternates in sign.  Look at the (-1).  It will alternate between positive and negative when k is odd or even.
-2 + 4 - 6 + 8 - 10
3)  Notice we used a different letter.  Doesn't matter!
5 + 8 + 11 + 14 + 17 + 20 + 23
(note)  Did you notice this is an arithmetic sequence!  The formula given was an explicit formula!
4)  This series is easy to spot.  It's the sum of the squares from 1 to 9! 5)  Notice this is an arithmetic series. Must use the formula to find the explicit formula.  d = 4, first term is 5 and counting the terms, the eighth term is 33.
5 + (n - 1)4 = 5 + 4n -4 = 4n +1 6)   Notice that this series is geometric. r = -1/3, first term is 1 and the last term -1/243 is the 6th term!  Use the geometric formula!
1( -1/3)(n - 1) = (-1/3)(n - 1) The next section works with a special type of proof called Mathematical Induction.   Current quizaroo #  13b 1)  Find: a)  0
b)  infinity
c)  does not exist
d)  3/4
e)  1 2)  Find: a)  0

b)  1
c)  infinity
d)  does not exist
e)  p/2 3)  Find the sum of the infinite geometric series:  1 + 1/4 + 1/16 + 1/ 64 + . . .

a)  4/3
b)  3/4
c)  5/4
d)  1
e)  it diverges 4)  What is the interval of convergence for the infinite geometric series:  1 + (x - 2) + (x - 2)2 + . . .

a)  0 < x < 2
b)  1 < x < 3
c)  x < 3
d)  x < 2
e)  doesn't have an interval of convergence, since it always diverges 5)  Find the sum of the series: a)  60
b)  54
c)  58
d)  56
e)  15  