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Lindsay K.

what do you need to do if you have a telescoping sum?

write out the first and last few entries to see what does and doesn't cancel out

what is the formula for a geometrical series?

Σar^{i-1}

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what is the sum of Σ^{∞}ar^{n-1}?

a/1-r

what will a geometric series equal?

= a + ar + ar^{2} + ar^{3} +...+ ar^{n-1}

= a - ar^{n }(when multiplied by 1-r)

= (a/1-r)(1-r^{n})

what are the p-integral rules for series?

1/x^{P}

- if p ≤ 1, divergent (∞)

- if p > 1, convergent (#)

^{}

What are the first 2 type I integrals?

1) given f(x) is continuous on [a, ∞); ∫_{a}^{∞}f(x)dx = lim _{b→∞ }∫_{a}^{b}f(x)dx

2) given f(x) is continuous on (∞, b]; ∫_{-∞}^{b}f(x)dx = lim_{a→-∞ }∫_{a}^{b}f(x)dx

what is the last type of improper integral type I?

3) given f(x) is continuous on (-∞,∞);

∫_{-∞}^{∞}f(x)dx = ∫_{0}^{∞}f(x)dx + ∫_{-∞}^{0}f(x)dx

what are the first two types of type II improper integrals?

1) if f(x) is discontinuous at x=a; ∫_{a}^{b}f(x)dx = lim_{c→a+ }∫_{c}^{b}f(x)dx

2) if f(x) is discontinuous at x=b; ∫_{a}^{b}f(x)dx = lim_{c→b-} ∫_{a}^{c}f(x)dx

what is the last type of improper integral type II?

3) if f(x) is discontinuous at x=a&b;

∫_{a}^{b}f(x)dx = ∫_{a}^{c}f(x)dx + ∫_{c}^{b}f(x)dx = lim_{s→a+}∫_{s}^{c}f(x)dx + lim_{t→b- }∫_{c}^{t}f(x)dx

what are the p-integral rules for a sequence?

1/xP

- if p ≥ 1, divergent (∞)

- if p < 1, convergent (#)

what is the overall two rules of the comparison test?

1) if the larger one converges, so does the little one

2) if the smaller one diverges, so does the bigger one

what is included for explicit form

A_{n}

what is included in recursive form?

a_{n }and the first term

what are the two possible results of the limit of a sequence?

if the Limit exists, the sequence converges.

if the limit doesn't exist (0 or ∞) then the sequence diverges

what does it mean to be monotonic?

always decreases or alway increases

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what does it mean to be bounded?

can't pass a minimum or maximum

what does it mean if a function is both monotonic and bounded?

convergent

if An is defined recursively, what will you do?

show that it is monotonic and bounded

what are the possibilities for lim_{n→∞}ar^{n}? (3)

divergent if r>1

convergent if -1<r≤1

divergent if r≤-1

what are the possibilities for lim_{n→∞}(-1)^{n}b_{n}? (2)

divergent if bn→L≠0

convergent if bn→0

what is the special sum of Σ1?

n

what is the special sum of Σi?

n(n+1)/2

what is the special sum of Σi^{2}?

n(n+1)(2n+1)/6

what is the special sum of Σi^{3}?

n^{2}(n+1)^{2}/4

What does Σ(1/i)-(1/i+1) equal?

1-(1/n+1)

what is the equation for a telescoping sum?

Σ(1/i)-(1/i+1)

what two steps are there for induction? explain each step.

1) Base case - show that a1 is < or > a2

2) induction step - assume that at k it's still increasing ie. ak ≤ ak+1 and the step after that ie. ak+1 ≤ ak+2

using Series, how would you get a_{m}?

S_{m}-S_{m-1} = a_{m}

what is the divergence test?

if the limit of a_{m} goes to 0, the series is convergent, otherwise it's divergent

when is the sum of the series Σar^{n-1 }converging and diverging?

diverging when |r| ≥ 1

converging when |r| < 1

what is the integral comparison test for series?

if f(x) is a function and a_{n}=f(x) and f(x) is positive, continuous and decreasing, then both Σan and ∫f(x)dx converge - either both converge or diverge

what general equation do we use for error calculations?

S - S_{m} = R_{m}

what is the error comparison used for higher bounds?

S-S_{m} ≤ ∫_{m}^{∞}f(x)dx < R_{m}

what is the error comparison used for lower bounds?

S-S_{m} ≥ ∫_{m}^{∞}f(x)dx > R_{m}

what are the two parts to the series comparison test?

if an, bn ≥ 0;

if an ≤ bn, if bn converges so does an

if an ≥ bn, if bn diverges, so does an

what two things must happen for Σ(-1)^{n}b_{n to converge?}

go to zero (lim_{n→∞} b_{n} = 0) and decreases (b_{n} ≥ b_{n+1})

what does it mean for a series to be conditionally convergent?

|an| diverges but an converges

what does it mean for a series to be absolutely convergent?

of |an| is convergent (then an is also convergent)

what is the general formula for the ratio test?

lim_{n→∞}|a_{n+1}/a_{n}|

what are the possible outcomes of the ratio test? (3)

if the limit is:

> 1 divergent

= 1 test fails

< 1 absolutely converges

when do we use the root test?

when stuff is to "n" powers ie. (stuff)^{#n}

what are the three possible outcomes of the root test?

if the limit is

> 1, series diverges

= 1, test fails

< 1, series converges absolutely

what is the form of a direct comparison test?

where an > 0 and bn > 0; 0 < lim_{n→∞}an/bn < ∞

what are the possible outcomes of the direct comparison test?

An < Bn and if Σbn converges; convergent

Bn>An and if Σbn diverges; divergent

what happens to the determinant when you swap rows?

-det(A)

what happens to the determinant when you add a multiple of one row to another?

no change in determinant

what happens to the determinant when you multiply a row by a number?

k*det(A)

what is an alternate form of det(A)^{n}?

(detA)^{n}

what is an alternate form of det(A^{-1})?

1/(det(A))

what is det(A^{T})?

det(A) - no change

what is an alternate form of det(kA)?

K^{n}det(A) - n is a dimension

what is the 'new' formula for A^{-1}?

A^{-1} = ^{adjA}/_{detA}

what are the 6 parts to the mega-theorem?

1) A is non-singular (invertible) 2) A RREF is I 3)A can be expressed as a product of elementary matrices 4) Ax=b has a unique solution for any b 5) Ax=0 only has the trivial solution 6) Ax=b always has a solution

If A and B are square matrices of the same size, then what is det(AB)?

det(A)*det(B)

What is the characteristic equation of A? (eigenshit)

det(λI − A) = 0

If A is an n × n triangular matrix (upper triangular, lower triangular,

or diagonal), what are the eigenvalues?

the entries on the main diagonal of A

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