Flashcards in Series Deck (44):

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 nâ†’âˆž {a}=L

Equal to Lim nâ†’âˆž f(x)=L

3

## Converging sequences

### Has a Lim nâ†’âˆž {a}

4

## Diverging Sequences

### Has no Lim nâ†’âˆž {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}

12

## Harmonic Sequence

###
Increasing denominator value by one

Î£(1/k)=1+1/2+1/3+...

Limit of zero

13

## P-serries

###
Increasing denominator value by one with an exponent

Î£(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

16

## Root test

###
If p= Lim kâ†’âˆž k'dâˆš(a sub-k)

If 0

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 kâ†’âˆž for a/b is 0 and and be converge together

But

When Lim kâ†’âˆž for a/b is âˆž and and be diverge together

19

## Alternating harmonic serries

###
Harmonic function only the signs change with each term

Takes the form:

Î£[(-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 kâ†’âˆž 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 Î£f(a) becomes Î£|f(a)|

25

## Conditional Convergence

### When a series converges only when Î£f(a) but not for Î£|f(a)|

26

## Power Serries

###
Series of Exponentially increasing terms

Takes the form: Î£c*x^p

27

## Taylor serries

###
Series in the form Î£c(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 Î£c(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

44