Transcript Sequences and Series - wastudentscience.org

 Sequence: an
ordered list of numbers; the
numbers in this ordered list are called
"elements" or "terms“
 Series: the summation of the terms of a
sequence
 Ex) the arithmetic sequence: 1,2,3,4,5…
 Ex) the arithmetic series: 1+2+3+4+5…
A
sequence can be denoted as 𝐴𝑛
(pronounced “A sub n”) where 𝐴𝑖 is the ith
term of the sequence
 Series:
𝑆𝑛 , where 𝑆𝑖 is the sum of the first i
terms of the sequence defining the series
 Sigma
notation as well
 http://en.wikipedia.org/wiki/Summation#C
apital-sigma_notation
 Arithmetic
• Arithmetic series never converge
 Geometric
• Converging geometric series, when|r|<1

𝐴𝑛 = 𝐴1 + (𝑛 − 1)𝑑
• Where n is the term, 𝐴1 is the initial term, and d is the
“common difference” added each time

It is not possible to evaluate the infinite series
• Why?

We can find the sum over a finite interval
however
• Add the first term and last term, multiply this sum by the
number of terms, and divide by 2
 Why does this work?
 Geometric
 Where
Sequence: 𝐴𝑛 = 𝐴1 𝑟 𝑛−1
n is the term number, 𝐴1 is the
initial term, and r is the “common factor”
multiplied each time
Infinite Series: Evaluating the sum when |r| < 1
•
𝐴1
1−𝑟
Evaluating the sum when |r| > 1:
• Does not sum up to a finite value when infinitely many terms
are added – why?
Finite Series:
• The sum of n terms of the starting with 𝐴1 :
•
𝐴1 (1−𝑟 𝑛+1 )
1−𝑟
• Prove that these expressions are valid – start with the second one.
• Hint: Try using the same tactic as we did for the arithmetic series
(manipulate the sum of the series as a whole)
 If
the sum of the first ten terms of an
arithmetic progression is four times the
sum of the first five terms, find the ratio of
the first term to the common difference.
 If
the sum of the first ten terms of an
arithmetic progression is four times the
sum of the first five terms, find the ratio of
the first term to the common difference.
•
𝑎
𝑑
= 1/2

In a story called One Grain of Rice, a young girl
convinces a stingy king to give one grain of rice
on the first day, but then double the amount
given each day, and continue to do so for 30
days. (So the 2nd day, she would receive 2 grains,
on the 3rd, 4, and so on.) If there are 29,000 grains
in a pound, how many pounds would she receive
after 30 days?
2
3
2
9
2
2
2
2
+ –
….+
27
81 243
59049

Find the sum of 2 –

Express 0.545454545454…. as a fraction.
+
–

1= 0.99999999999999….
• What…? Why?

Lalith and Oscar decide to play a game involving
dice. Lalith begins the game by rolling a single die,
and wins if he rolls a 3 or a 5. If Lalith doesn’t win, he
passes the die to Oscar, who wins if he rolls an even
number. If Oscar doesn’t win on his turn in this round,
he passes the die back to Lalith, and the cycle
continues. The game continues in this fashion until
somebody wins. If Oscar and Lalith decide to play for
money, and Oscar must pay Lalith $10 if Lalith wins,
what should Lalith pay Oscar when he wins in order
to ensure that the game is fair?
 In
this case, it’s easier to show an example
rather than give a definition. Partial fractions
are necessary for evaluating these.

1
∞
𝑘=1 𝑘 2 +𝑘
• The above series is equivalent to
∞ 1
𝑘=1 𝑘
−
1
. Why?
𝑘+1
• If we evaluate this infinite series beginning at k = 1,
what is the value?
 Determine
the infinite summation if each
sequence begins at k =1 for:

1
∞
𝑘=1 4𝑘 2 −1

∞ 2𝑘−1
𝑘=1 3𝑘+1