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

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