MATH 152 Exam III Formula Recognition

sin x Maclaurin series

sin x = sum from n=0 to infinity of (-1)^n x^(2n+1)/(2n+1)!

cos x Maclaurin series

cos x = sum from n=0 to infinity of (-1)^n x^(2n)/(2n)!

e^x Maclaurin series

e^x = sum from n=0 to infinity of x^n/n!

Geometric series

1/(1-x) = sum from n=0 to infinity of x^n, valid for |x| < 1

ln(1+x) series

ln(1+x) = sum from n=1 to infinity of (-1)^(n+1) x^n/n

arctan x series

arctan x = sum from n=0 to infinity of (-1)^n x^(2n+1)/(2n+1)

Sine vs arctan cue

Both alternate and use odd powers, but sine has factorial (2n+1)! while arctan has only (2n+1).

Series recognition: factorial in denominator

Think e^x, sin x, or cos x depending on powers/signs.

Series recognition: no factorial, plain powers

Think geometric series or ln/arctan depending on denominator and powers.

Substitution in a known series

Replace every x in the base series with g(x), then simplify only if needed.

Arctan disguise pattern

sum (-1)^n a^(2n+1)/(b^(2n)(2n+1)) = b arctan(a/b)

Power series interval first step

Rewrite into inside^n form, solve |inside| < 1, then check endpoints separately.

Endpoint: alternating 1/sqrt(n)

sum (-1)^n/sqrt(n) converges by Alternating Series Test.

Endpoint: positive 1/sqrt(n)

sum 1/sqrt(n) diverges because it is p-series with p = 1/2 < 1.

Alternating Series Test conditions

a_n decreases and a_n approaches 0.

Absolute value inequality rule

|A| < c means -c < A < c.

Binomial series formula

(1+x)^p = sum from n=0 to infinity of binomial(p,n) x^n

Generalized binomial coefficient

binomial(p,n) = p(p-1)(p-2)…(p-n+1)/n!

Binomial series first terms

(1+x)^p = 1 + px + p(p-1)x^2/2! + p(p-1)(p-2)x^3/3! + …

Binomial series convergence

For non-integer p, use |x| < 1 first, then check endpoints separately.

Binomial substitution cue

If the expression is (1 + something)^p, plug that something in for x.

Parabola horizontal form

(y-k)^2 = 4p(x-h)

Parabola vertical form

(x-h)^2 = 4p(y-k)

Parabola vertex

Vertex is (h,k).

Parabola: y squared cue

If y is squared, the parabola opens left or right.

Parabola: x squared cue

If x is squared, the parabola opens up or down.

Horizontal parabola focus and directrix

For (y-k)^2 = 4p(x-h): focus (h+p,k), directrix x = h-p.

Vertical parabola focus and directrix

For (x-h)^2 = 4p(y-k): focus (h,k+p), directrix y = k-p.

Completing the square rule

For y^2 + by, add and subtract (b/2)^2.

Ellipse cue

Both x^2 and y^2 appear, same sign, sum equals 1.

Horizontal ellipse form

(x-h)^2/a^2 + (y-k)^2/b^2 = 1, bigger denominator under x.

Vertical ellipse form

(x-h)^2/b^2 + (y-k)^2/a^2 = 1, bigger denominator under y.

Ellipse center

Center is (h,k).

Ellipse vertices

Move a units from the center along the major axis.

Ellipse foci relationship

c^2 = a^2 - b^2.

Ellipse direction cue

Bigger denominator tells the major axis direction.

Hyperbola cue

Both x^2 and y^2 appear with opposite signs.

Horizontal hyperbola form

(x-h)^2/a^2 - (y-k)^2/b^2 = 1.

Vertical hyperbola form

(y-k)^2/a^2 - (x-h)^2/b^2 = 1.

Hyperbola direction cue

The positive fraction tells the opening direction.

Hyperbola vertices

Move a units from the center in the opening direction.

Hyperbola foci relationship

c^2 = a^2 + b^2.

Horizontal hyperbola asymptotes

y-k = plus/minus (b/a)(x-h).

Vertical hyperbola asymptotes

y-k = plus/minus (a/b)(x-h).

Ellipse vs hyperbola c formula

Ellipse: c^2 = a^2 - b^2. Hyperbola: c^2 = a^2 + b^2.

Conic quick classifier

Parabola has one squared variable; ellipse has same-sign squared terms; hyperbola has opposite-sign squared terms.