MATH1051 Final

what is the p-series test?

a series is convergent if p > 1 and divergent if p <-= 1

what is the divergence test/nth term test?

if the limit of a_n =/ 0, then the series diverges

what is the geometric series?

the series arⁿ is convergent if |r| < 1 with the series = a/1-r and divergent if |r| >= 1

what is the comparision test?

if series b_n converges and 0 <= a_n <= b_n, then an converges OR if b_n diverges and a_n >= b_n, an diverges

what is the alternating series test?

when a series alternates in signs:

if a_n = (-1)ⁿ b_n, then:

1. lim b_n = 0

2. b_n+1 <= b_n for all n >=0

then series is convergent

what is absolute convergence?

if the series |a_n| converges

what is conditionally convergent?

a_n converges but NOT absolutely

when does a series converge by ratio test?

if lim |a_n+1/a_n| = L < 1, series is absolutely convergent

when does a series diverge by ratio test?

if lim |a_n+1/a_n| = L > 1, series is divergent

when is the ratio test inconclusive?

if lim |a_n+1/a_n| = 1 or not defined

what is a power series?

series of C_n (x-a)ⁿ which = C₀ + C₁(x-a) + C₂ (x-a)² …

when does a series converge using radius of convergence?

1. x = a (r = 0)

2. x = all real numbers (r = infinity)

3. if |x-a| < r , the series converges and if |x-a| > r, series diverges

what is the taylor series for cos x ?

1 - x²/2! + x⁴/4! - x⁶/6! +…

what is the taylor series for sin x ?

1 - x³/3! + x⁵/5! - x⁷/7! +…

what is the taylor series for e^x?

1 + x + x²/2! + x³/3! +…

what happens when a function is on the boundaries for an integral?

1. let u = the boundary function

2. make a new function in terms of u e.g H(u)

3. find the derivative of the original function by deriving dH(u) x du/dx

what is the formula for integration by parts?

integral of uv’ = uv - integral of vu’ → following LIATE

what is the formula for the dot product?

||v|| . ||w|| cos theta or v . w = v1w1 + v2w2…

what is the formula for a parallelpiped?

V = | u . (v x w)|

what is vector projection?

projecting w onto v = proj_v w

= (w.v)/||v||² x v

how to determine the cross product using determinants?

have a 3×3 matrix with i j k along first row and vector v and w along next two rows and use co-factor expansion (j is -ve)

what is matrix transposing?

A^T where the rows become columns and vice versa

what is the rule for matrix transposed expansion?

(AB)^T = B^TA^T

how do you solve a gaussian problem?

after solving row operations:

1. write out all the equations

2. if there’s a free variable, get all other variables in term of that free variable alpha

3. write out S = {()} in alpha and put into line form

what is the nullspace?

Ax = 0 where in a matrix, it is 0’s along the bottom row and 0’s as the answer column

if Ax = b exists and NS(A) = {0}, Ax = b is unique

what does it mean by singular?

no inverse → determinant = 0

what is the formula for an inverse?

A^-1 = 1/det[d -b -c a]

how to determine the inverse using gaussian?

AX = I so set up two gaussian columns [A] and [I] and solve until [I] | [B] where B is A^-1

what is cofactor expansion?

C_ij = (-1)^i+j M_ij

where M is delete the i column and row j

what is cofactor expansion for a 3×3?

det = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ or along any row/column

what happens when row/columns are swapped for determinants?

det is -ve

what is the determinant of lower/upper triangular matrices?

det = product of main diagonal

what is the determinant of a transposed matrix?

the same as not transposed matrix

what is the eigen formula?

Ax = eigen x where x is eigenvector

what is a formula for eigenvectors and determinants?

(A - eigen I) x = 0 where the determinant of |A- eigen I| = 0

what is a characteristic equation?

using |A- eigen I| = 0 to find eigenvalues of A

→ should expect the matrix to be 00|0 / free variables

what is linear independence?

no vectors in the set can be expressed as a linear combination of the others → coefficient of all = 0

what is linear dependence?

at least one vector in the set can be expressed as a linear combination of the others, meaning not all coefficients = 0

how to test for linear independence?

a₁₍A) + a₂(B) = 0 vector where a₁ and a₂ should = 0

how to test for linear dependence?

set up the equation a₁(A) + a₂(B) = 0 vector, and try and put into Ax = 0 form

when are two vectors linearly dependent?

when one of them is a multiple of the other

what is a vector space?

when a subset V satisfies:

0. V is non-empty (must have 0 vector)

1. if v is an element of V and w is an element of V when v + w is an element of V (closure under addition)

2. if v is an element of V then alpha v is an element of V (closure under scalar multiplication)

how to determine vector spaces?

0. check for the 0 vector

1. use v and w and use LHS of equation to find

2. add a scalar to find if alpha is in V

what is a telescoping series?

a series where most terms cancel out when expanded, resulting in a simple expression for the sum

how to determine if telescoping series is convergent?

if limit of n partial sum exists then the series exists

what is the partial sum Sn?

The partial sum $S_n$ of a series is the sum of the first $n$ terms of the series e.g S₀ = S₀, S₁ = S₀ + S₁