Flashcards in Serries Deck (44):

0

## Sequence

###
Ordered list of numbers

ie {2, 4, 6, 8, 10, ...}

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## Recurrance relation

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Function that describes the increasing values of a sequence

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## Limit of a sequence

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The limit as the number of terms increases towards infinity

Lim n→∞ {a}=L

Equal to Lim n→∞ f(x)=L

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## Converging sequences

### Has a Lim n→∞ {a}

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## Diverging Sequences

### Has no Lim n→∞ {a}

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## Infinite Serries

### A serries with an infinite number if terms

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## Nondecreasing Sequences

### Each term of the sequence increases

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## Monotonic Sequences

### Series in which the terms neither continuously increase or decrease

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## Bounded serries

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

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## Geometric Sequence

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

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## Sequence Ratio

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

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## Squeeze Theorem for sequences

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If {a}<{c}

And Lim{a}=Lim{c}

Then Lim{b} is equal too

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## Harmonic Sequence

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Increasing denominator value by one

Σ(1/k)=1+1/2+1/3+...

Limit of zero

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## P-serries

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Increasing denominator value by one with an exponent

Σ(1/k^p)

Limit of zero

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## Convergence Test

### Sequence converges if the sequence limit equals zero

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## Ratio Test

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

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## Root test

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If p= Lim k→∞ k'd√(a sub-k)

If 0

<1, the sequence converges

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## Comparison Test

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If all the terms of series1 are greater than the terms of series2

They either converge together or diverge together

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## Limit comparison test

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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

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## Alternating harmonic serries

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Harmonic function only the signs change with each term

Takes the form:

Σ[(-1)^(k+1)]/k

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## Alternating serries

### Series in which the terms alternate between positive and negative

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## Nonincreaseing

### Each term of the series decreses

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## Alternating Series test

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

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## Series Remainder

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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

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## Absolute convergence

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

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## Conditional Convergence

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

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## Power Serries

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Series of Exponentially increasing terms

Takes the form: Σc*x^p

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## Taylor serries

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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!

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## Taylor's Theorem

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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

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## Interval of convergence

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

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## The radius of convergence

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

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## Power series center

### The 'a' value in Σc(x-a)^k

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## Maclaurin Series

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Any Taylor Series centered at 0

Meaning the a-value is zero

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## Linear Term (for linear aproximation series)

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The portion of the series sum that takes the form:

f(a)+f'(a)(x-a)

Equal to p1(x)

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## Quadratic term (for quadratic approximation)

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The portion of the series sum that takes the form:

C(x-a)^2

Always at the very end

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## n-th Taylor Polynomial

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Denoted pn

Has a center at 'a'

Takes the form:

Pn=f(a)+f'(a)(x-a)+...+(nth-f(a)/n!)(x-a)^n

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## Differentiating a series

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Find the polynomial

Differentiate one term at a time

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## Integrating any serries

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Find the polynomial

Integrate one term at a time

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## Binomial Coefficients

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Written as (p over k)

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

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## Binomial serries

### Series in which each term is a binomial coefficient

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## Convergence of the Series

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

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## Whys the taylor series so important?

### Describes any function

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## Differentiating or integrating a power series

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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

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