Series and Polar Coordinates (BC Only)

BC Only: Testing for Convergence/Divergence of a Series

Sequence of Partial Sums

Given the series (∑_^)((a_n))=(a_1)+(a_2)+(a_3)+⋯

The sequence of partial sums for the series is

(S_1)=(a_1), (S_2)=(a_1)+(a_2), (S_3)=(a_1)+(a_2)+(a_3) ⋯ (S_n)=(a_1)+(a_2)+(a_3)+⋯+(a_n)

Nth Term

If the terms of a sequence do not converge to 0, then the series must diverge.

If (lim_n→∞)((a_n))≠0, then (∑_^)((a_n)) diverges

If (lim_n→∞)((a_n))=0, then the test is inconclusive

P - Series

The form of a p - series is

(∑_^)(1/(np))

If p>1, then the series converges

If p<1, then the series diverges

Geometric Series

A geometric series is any series of the form

(∑_n=0^∞)(a*rn)

If |r|<1, then the series converges to a/(1-r), Series must be indexed at n=0

If |r|>1, then the series diverges

Telescoping Series

A telescoping series is any series of the form

(∑_^)((a_n))-(a_n+1)

Convergence and divergence is found using a sequence of partial sums

Partial decomposition may be used to break a single rational series into the difference of two series that form the telescoping series.

Integral

If ƒ is positive, continuous, and decreasing for x≥1, then

(∑_n=1^∞)((a_n)) and (∫_1^∞)(ƒ(x)*d(x))

either both converge or both diverge

Alternating Series

A series, containing both positive terms, negative terms, and (a_n)>0, of the form

(∑_n=1^∞)((-1)n*(a_n)) or (∑_n=1^∞)((-1)(n+1)*(a_n))

The series’ converge if both of the following conditions are met

(a_n+1)≤(a_n) for all n

(lim_n→∞)((a_n))=0

Direct Comparison

When comparing two series, if (a_n)≤(b_n) for all n,

If (∑_^)((a_n)) diverges, then (∑_^)((b_n)) diverges

If (∑_^)((b_n)) converges, then (∑_^)((a_n)) converges

The convergence or divergence of the series chosen for comparison should be known

Limit Comparison

If (a_n)>0 and (b_n)>0 and (lim_n→∞)((a_n)/(b_n))=L, where L is finite and positive, then the series (∑_^)((a_n)) and (∑_^)((b_n)) either both converge or both diverge

The convergence or divergence of the series chosen for comparison should be known

When choosing a series to compare to, disregard all but the highest powers (growth factor) in the numerator and denominator

Root

Given a series (∑_^)((a_n))

If (lim_n→∞)(√(n,|(a_n)|))<1, then (∑_^)((a_n)) converges

If (lim_n→∞)(√(n,|(a_n)|))>1, then (∑_^)((a_n)) diverges

If (lim_n→∞)(√(n,|(a_n)|))=1, then the root test is inconclusive

This is a test for absolute convergence

Ratio

Given a series (∑_^)((a_n))

If (lim_n→∞)(|(a_n+1)/(a_n)|)<1, then (∑_^)((a_n)) converges

If (lim_n→∞)(|(a_n+1)/(a_n)|)>1, then (∑_^)((a_n)) diverges

If (lim_n→∞)(|(a_n+1)/(a_n)|)=1, then the ratio test is inconclusive.

This is a test for absolute convergence

BC Only: Absolute vs Conditional Convergence

For a series (∑_^)((a_n)) with both positive and negative terms

If (∑_n=1^∞)(|(a_n)|) converges, then (∑_n=1^∞)((a_n)) also converges. (∑_n=1^∞)((a_n)) is said to be absolutely convergent

If (∑_n=1^∞)(|(a_n)|) diverges, but (∑_n=1^∞)((a_n)) converges, (∑_n=1^∞)((a_n)) is said to be conditionally convergent

BC Only: Alternating Series Remainder Theorem

Given (∑_^)((a_n)) is a convergent alternating series, the error associated with approximating the sum of the series by the first n terms is less than or equal to the first omitted term.

(∑_n=1^∞)((-1)(n+1)*(a_n))=S⋍(S_n)=(a_1)-(a_2)+⋯+(-1)(n+1)*(a_n)

Error =|S-(S_n)|≤|(a_n+1)|

BC Only: Power Series

Power Series Structure and Characteristics

Power Series centered at x=0

(∑_n=0^∞)((a_n)*xn)=(a_0)+(a_1)*x+(a_2)*x2+⋯+(a_n)*xn+⋯

Power Series centered at x=c

(∑_n=0^∞)((a_n)(x-c)n)=(a_0)+(a_1)(x-c)+(a_2)(x-c)2+⋯+(a_n)(x-c)n+⋯

A function f can be represented by a power series , where the power series converges to the function in one of three ways:

The power series only converges at the center x=c

The power series converges for all real values of x

The power series converges for some interval of values such that |x-c|<R, where R is the radius of convergence of the power series.

Interval of Convergence: Find this by applying the Ratio to the given series

If R=0, then the series converges only at x=c

If R=∞, then the series converges for all real values of x

If the Ratio Test results in an expression of the form |x-c|<R, then the interval of convergence is of the form c-R<x<c+R

The convergence at the endpoints of the interval of convergence should be tested separately

BC Only: Taylor and Maclaurin Series (specific power series)

If a function of ƒ has derivatives of all orders at x=c, then the series is called a Taylor Series for ƒ centered at c. A Taylor series centered at 0 is also known as a Maclaurin Series.

Maclaurin Series:

ƒ(x)=ƒ(0)+(ƒ^′)(0)*x+((ƒ^″)(0)*x2)/2!+((ƒ^″*′)(0)*x3)/3!+⋯+(ƒn*(0)*xn)/n!+⋯=(∑_n=0^∞)((ƒn*(0)*xn)/n!)

Taylor Series:

ƒ(x)=ƒ(c)+(ƒ^′)(c)*(x-c)+((ƒ^″)(c)*(x-c)2)/2!+((ƒ^″*′)(c)*(x-c)3)/3!+⋯+(ƒn*(c)*(x-c)n)/n!+⋯

=(∑_n=0^∞)((ƒn*(0)*xn)/n!)

BC Only: Lagrange Remainder of a Taylor Polynomial

When approximating a function ƒ(x) using an nth degree Taylor polynomial, (P_n)(x), the associated error, (R_n)(x), is bounded by

|(R_n)(x)|=|ƒ(x)-(P_n)(x)|≤|((x-c)(n+1))/(n+1)!⋅max(ƒ(n+1)*(z))| where c≤z≤x

BC Only: Polar Coordinates

The polar coordinates (r,θ) of a point are related to the rectangular coordinates (x,y) as follows:

x=r*cos(θ)

y=r*sin(θ)

r2=x2+y2

tan(θ)=x/y

If ƒ is a differentiable function of θ (smooth curve), then the slope of the line tangent to the graph of r=ƒ(θ) at the point (r,θ) is

d(y)/d(x)=d(y)/d(θ)/d(x)/d(θ)=((r^′)*sin(θ)+r*cos(θ))/((r^′)*cos(θ)-r*sin(θ))=((ƒ^′)(θ)*sin(θ)+ƒ(θ)*cos(θ))/((ƒ^′)(θ)*cos(θ)-(ƒ^′)(θ)*sin(θ))

If r=ƒ(θ) is a smooth curve on the interval [α,β], where α and β are radial lines, then the area enclosed by the graph is

Area =1/2*(∫_α^β)(r2*d(θ))=1/2*(∫_α^β)([ƒ(θ)]2*d(θ))