Series Flashcards
Recurrance relation
Function that describes the increasing values of a sequence
Limit of a sequence
The limit as the number of terms increases towards infinity
Lim n→∞ {a}=L
Equal to Lim n→∞ f(x)=L
Converging sequences
Has a Lim n→∞ {a}
Diverging Sequences
Has no Lim n→∞ {a}
Infinite Serries
A serries with an infinite number if terms
Nondecreasing Sequences
Each term of the sequence increases
Monotonic Sequences
Series in which the terms neither continuously increase or decrease
Bounded serries
A series whose terms are all less than or equal to a finite number
Geometric Sequence
Series in which the last term is multiplied by an unchanging number
Sequence Ratio
The unchanging number by which the terms in geometric sequence are multiplied
Squeeze Theorem for sequences
If {a}
Harmonic Sequence
Increasing denominator value by one
Σ(1/k)=1+1/2+1/3+…
Limit of zero
P-serries
Increasing denominator value by one with an exponent
Σ(1/k^p)
Limit of zero
Convergence Test
Sequence converges if the sequence limit equals zero
Ratio Test
If the ratio ‘r’ is 0
Root test
If p= Lim k→∞ k’d√(a sub-k)
If 0<p></p>
Comparison Test
If all the terms of series1 are greater than the terms of series2
They either converge together or diverge together
Limit comparison test
When Lim k→∞ for a/b is 0 and and be converge together
But
When Lim k→∞ for a/b is ∞ and and be diverge together
Alternating harmonic serries
Harmonic function only the signs change with each term
Takes the form:
Σ[(-1)^(k+1)]/k
Alternating serries
Series in which the terms alternate between positive and negative
Nonincreaseing
Each term of the series decreses
Alternating Series test
An alternating series converges if Lim k→∞ a=0
Series Remainder
Rn=|S-Sn|
The absolute error in approximating the value to which an infinite series converges, using the convergent value at the n-the term as the measurement
Absolute convergence
When a series still converges even when the Σf(a) becomes Σ|f(a)|
Conditional Convergence
When a series converges only when Σf(a) but not for Σ|f(a)|
Power Serries
Series of Exponentially increasing terms
Takes the form: Σc*x^p
Taylor serries
Series in the form Σc(x-a)^k
Each coefficient takes the form:
k-th derivative of the function of a over k!
[f^k(a)]/k!
Taylor’s Theorem
The function f(x) output is equal to the n-th output, plus the remainder Rn
f(x)=pn(x)+Rn(x)
Rn(x)=[f^(n+1)(c)]/[(n+1)!]*(x-a)^(n+1)
Need to write this out
Interval of convergence
The set of x-values on which the power series converges
The radius of convergence
Distance from the center of the series to the boundary of the interval
Power series center
The ‘a’ value in Σc(x-a)^k
Maclaurin Series
Any Taylor Series centered at 0
Meaning the a-value is zero
Linear Term (for linear aproximation series)
The portion of the series sum that takes the form:
f(a)+f’(a)(x-a)
Equal to p1(x)
Quadratic term (for quadratic approximation)
The portion of the series sum that takes the form:
C(x-a)^2
Always at the very end
n-th Taylor Polynomial
Denoted pn
Has a center at ‘a’
Takes the form:
Pn=f(a)+f’(a)(x-a)+…+(nth-f(a)/n!)(x-a)^n
Differentiating a series
Find the polynomial
Differentiate one term at a time
Integrating any serries
Find the polynomial
Integrate one term at a time
Binomial Coefficients
Written as (p over k) (P(p-1)(p-2)...(p-k+1))/k!
Binomial serries
Series in which each term is a binomial coefficient
Convergence of the Series
Rn(x)=(n-th f(c))/(n+1)! (x-a)^(n+1)
Whys the taylor series so important?
Describes any function
Differentiating or integrating a power series
Find the maclaurin series for the function in question (or vice versa)
Limit that series to the interval
Calculate the integral or derivative for each term
Finding the power series of a function
Find the interval of convergence
Substitute a function within that interval
…See book for details
Sequence
Ordered list of numbers
ie {2, 4, 6, 8, 10, …}
