Sequences and Series Flashcards
Arithmetic sum:
Sn = 0.5n(2a+(n-1)d)
= 0.5n(a+l)
Geometric sum:
Sn = a(1-r^n)/(1-r)
Arithmetic term:
a + (n-1)d
Geometric term:
ar^n-1
Sum to infinity:
a/1-r
Two series might be interleaved so
find the sum of each one and addd
For oscillating series between positive and negative
find the running total
A typical trick is to replace xn+1 with
the expressions involving xn and yn, then simplifying
Multiple series interleaved
find sum of interleaved
Oscillating positive and negative
Consider the running total
(n+1)th term
replace n with n+1`
A typical trick is to replace xn+1 with the expressions involving xn and yn
Sometimes you can reapply your recurrence relationship to obtain larger values needed
Just writing xk+1 in two different ways here (in terms of both Ak and Ak+1 for example) and then comparing coefficients, will do the trick.
We’re comparing parameters from the current and next term in the sequence
Recurrence relationships
compare coefficients
If n>= a,
Sub in a to make series simpler, or the simplest number allowed
look for multiple series’ in a single series
find sum of each
Find patterns in function sequences,
whys this true
When cannot find series formula
check that all iterations are correct
recurrence:
relate xn+1 to xn, try values
Try and relate separate recurrence formula to each other
series find patterns
xn > xn+1 –>
n^2 + 2n > (n+1)^2 + 2(n+1)
f(n) series… patterns
Recurrence formula: compare coefficients
List results find pattern between n and f(n) but
check that all iterations are correct
Oscillating series
split into two
Finding a formula from recurrence relationship: list answers
Use recurrence relationships
Base cases, n be as small as possible in order to simplify problem
x = a + (n-1)d
∑n,r=1 (r) =
n(n+1)/2
∑n,r=1 (r^2) =
(2n+1)(n+1)/6
∑n,r=1 (r^3) =
0.25n^2(n+1)^2
∑n,r=1 (k) =
kn
∑n,r=1 (ar+b) =
bn + a∑n,r=1 (r)
∑n,r=k (r) =
∑n,r=1 (r) - ∑k-1,r=1 (r)
