Serries

Question 0

Recurrance relation

Answer 0

Function that describes the increasing values of a sequence

Question 1

Sequence

Answer 1

Ordered list of numbers

ie {2, 4, 6, 8, 10, …}

Question 2

Limit of a sequence

Answer 2

The limit as the number of terms increases towards infinity
Lim n→∞ {a}=L
Equal to Lim n→∞ f(x)=L

Question 3

Converging sequences

Answer 3

Has a Lim n→∞ {a}

Question 4

Diverging Sequences

Answer 4

Has no Lim n→∞ {a}

Question 5

Infinite Serries

Answer 5

A serries with an infinite number if terms

Question 6

Nondecreasing Sequences

Answer 6

Each term of the sequence increases

Question 7

Monotonic Sequences

Answer 7

Series in which the terms neither continuously increase or decrease

Question 8

Bounded serries

Answer 8

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

Question 9

Geometric Sequence

Answer 9

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

Question 10

Sequence Ratio

Answer 10

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

Question 11

Squeeze Theorem for sequences

Answer 11

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

Question 12

Harmonic Sequence

Answer 12

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

Question 13

P-serries

Answer 13

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

Question 14

Convergence Test

Answer 14

Sequence converges if the sequence limit equals zero

Question 15

Ratio Test

Answer 15

If the ratio ‘r’ is 0<1, the sequence converges

Question 16

Root test

Answer 16

If p= Lim k→∞ k’d√(a sub-k)

If 0<p><1, the sequence converges

Question 17

Comparison Test

Answer 17

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

Question 18

Limit comparison test

Answer 18

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

Question 19

Alternating harmonic serries

Answer 19

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

Question 20

Alternating serries

Answer 20

Series in which the terms alternate between positive and negative

Question 21

Nonincreaseing

Answer 21

Each term of the series decreses

Question 22

Alternating Series test

Answer 22

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

Question 23

Series Remainder

Answer 23

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

Question 24

Absolute convergence

Answer 24

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

Question 25

Conditional Convergence

Answer 25

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

Question 26

Power Serries

Answer 26

Series of Exponentially increasing terms

Takes the form: Σc*x^p

Question 27

Taylor serries

Answer 27

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!

Question 28

Taylor’s Theorem

Answer 28

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

Question 29

Interval of convergence

Answer 29

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

Question 30

The radius of convergence

Answer 30

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

Question 31

Power series center

Answer 31

The ‘a’ value in Σc(x-a)^k

Question 32

Maclaurin Series

Answer 32

Any Taylor Series centered at 0

Meaning the a-value is zero

Question 33

Linear Term (for linear aproximation series)

Answer 33

The portion of the series sum that takes the form:
f(a)+f’(a)(x-a)
Equal to p1(x)

Question 34

Quadratic term (for quadratic approximation)

Answer 34

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

Question 35

n-th Taylor Polynomial

Answer 35

Denoted pn
Has a center at ‘a’

Takes the form:
Pn=f(a)+f’(a)(x-a)+…+(nth-f(a)/n!)(x-a)^n

Question 36

Differentiating a series

Answer 36

Find the polynomial

Differentiate one term at a time

Question 37

Integrating any serries

Answer 37

Find the polynomial

Integrate one term at a time

Question 38

Binomial Coefficients

Answer 38

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

Question 39

Binomial serries

Answer 39

Series in which each term is a binomial coefficient

Question 40

Convergence of the Series

Answer 40

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

Question 41

Whys the taylor series so important?

Answer 41

Describes any function

Question 42

Differentiating or integrating a power series

Answer 42

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

Question 43

Finding the power series of a function

Answer 43

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