Serries Flashcards Preview

B) Calculus > Serries > Flashcards

Flashcards in Serries Deck (44):
0

Sequence

Ordered list of numbers
ie {2, 4, 6, 8, 10, ...}

1

Recurrance relation

Function that describes the increasing values of a sequence

2

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

3

Converging sequences

Has a Lim n→∞ {a}

4

Diverging Sequences

Has no Lim n→∞ {a}

5

Infinite Serries

A serries with an infinite number if terms

6

Nondecreasing Sequences

Each term of the sequence increases

7

Monotonic Sequences

Series in which the terms neither continuously increase or decrease

8

Bounded serries

A series whose terms are all less than or equal to a finite number

9

Geometric Sequence

Series in which the last term is multiplied by an unchanging number

10

Sequence Ratio

The unchanging number by which the terms in geometric sequence are multiplied

11

Squeeze Theorem for sequences

If {a}<{c}
And Lim{a}=Lim{c}
Then Lim{b} is equal too

12

Harmonic Sequence

Increasing denominator value by one
Σ(1/k)=1+1/2+1/3+...
Limit of zero

13

P-serries

Increasing denominator value by one with an exponent
Σ(1/k^p)
Limit of zero

14

Convergence Test

Sequence converges if the sequence limit equals zero

15

Ratio Test

If the ratio 'r' is 0<1, the sequence converges

16

Root test

If p= Lim k→∞ k'd√(a sub-k)
If 0

<1, the sequence converges

17

Comparison Test

If all the terms of series1 are greater than the terms of series2
They either converge together or diverge together

18

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

19

Alternating harmonic serries

Harmonic function only the signs change with each term
Takes the form:
Σ[(-1)^(k+1)]/k

20

Alternating serries

Series in which the terms alternate between positive and negative

21

Nonincreaseing

Each term of the series decreses

22

Alternating Series test

An alternating series converges if Lim k→∞ a=0

23

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

24

Absolute convergence

When a series still converges even when the Σf(a) becomes Σ|f(a)|

25

Conditional Convergence

When a series converges only when Σf(a) but not for Σ|f(a)|

26

Power Serries

Series of Exponentially increasing terms
Takes the form: Σc*x^p

27

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!

28

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

29

Interval of convergence

The set of x-values on which the power series converges

30

The radius of convergence

Distance from the center of the series to the boundary of the interval

31

Power series center

The 'a' value in Σc(x-a)^k

32

Maclaurin Series

Any Taylor Series centered at 0
Meaning the a-value is zero

33

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)

34

Quadratic term (for quadratic approximation)

The portion of the series sum that takes the form:
C(x-a)^2
Always at the very end

35

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

36

Differentiating a series

Find the polynomial
Differentiate one term at a time

37

Integrating any serries

Find the polynomial
Integrate one term at a time

38

Binomial Coefficients

Written as (p over k)
(P(p-1)(p-2)...(p-k+1))/k!

39

Binomial serries

Series in which each term is a binomial coefficient

40

Convergence of the Series

Rn(x)=(n-th f(c))/(n+1)! (x-a)^(n+1)

41

Whys the taylor series so important?

Describes any function

42

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

43

Finding the power series of a function

Find the interval of convergence
Substitute a function within that interval
...See book for details