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.
Snoopy says, " Let's fly into that sigma stuff!"
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
Ernie says, "Stick with
me and I will give you the answers! Look below and all will be known!"
Answers to above problems:
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.
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
Snoopy says, " Let's fly into that sigma stuff!"
Ernie says, "Stick with
me and I will give you the answers! Look below and all will be known!"
