Sequences and Series Review
geometric series formula
Sn=a1(1-r^n)/1-r READ: the sum of n terms of a geometric series is equal to the first term multiplied by 1 minus the common ratio raised to the n power all divided by 1 minus the common ratio.
arithmetic series formula
Sn=n[(a1+an)/2] READ: The sum of n terms of an arithmetic series is equal to the number of terms multiplied by the average of the first term and the last term.
Two things that must be given for a recursive formula:
1: first term 2: pattern rule
All infinite arithmetic series _________
DIVERGE
convergent series formula
S=a1/(1-r) READ: The sum of the infinite series is equal to the first term divided by 1 minus the common ratio.
What three things must be true for you to be able to determine the sum of an infinite series?
The series must be: 1. geometric 2. infinite 3. convergent
typical element
a general explicit formula for evaluating a specific term in a given series
arithmetic sequence
a sequence in which each term is found by adding or subtracting the same number from the previous term
geometric sequence
a sequence in which each term is found by multiplying the previous term by the same number
sequence
an ordered list of numbers 2, 4, 6, 8
series
an ordered sum of numbers 2+4+6+8
formula used to evaluate a term in an arithmetic sequence or series
an=a1+d(n-1) READ: Term n is equal to the first term plus the common difference times n minus 1.
formula used to evaluate a term in a geometric sequence or series
an=a1xr^n-1 READ: Term n is equal to the first term times the common ratio raised to the n-1 power. NOTE: Only the common ratio is raised to the n-1 power, not the entire right side of the formula.
common difference notation
d
upper bound
end term of a series summation
What function familay are geometric sequences and series related to?
exponential functions
"a sub 1"
first term of the sequence or series
What do you underline when annotating?
important/signal words NOTE: this includes things like units
What function family are arithmetic sequences and series related to?
linear functions
What do you circle when annotating?
numbers and number words
common ratio notation
r
index
start term of a series summation
sigma
sum NOTE: Sigma should automatically signal to you that you must find a sum. Sigma is only used in reference to series.
The sum of a divergent series is _________
sum does not exist OR sum DNE
A series diverges if _________
the absolute value of the common ratio is greater than 1
A series is convergent if _________
the absolute value of the common ratio is less than 1
common difference
the difference between consecutive terms of an arithmetic sequence
What do you box when annotating?
the question NOTE: this should tell you form your answer should take
common ratio
the ratio of successive terms of a geometric sequence
"a sub n" or "term n"
the term for which you are evaluating
n
the term number for which you are evaluating (sequences) OR the number of terms (series)