Putnam Exam Formulas/Concepts to Remember for the Exam last updated November 24, 2015

Probability and its applications in polynomial expansions

n Permutations: Sometimes denoted Pr or nP r, is an ordered arrangement of n distinct objects taken r at a time. The formula is n! P n = r (n − r)!

 n  Combinations: Sometimes denoted Cn or nCr or , means the number of combinations of r r n objects taken r at a time. The formula is

 n  n! P n = = r . r (n − r)!r! r!

Binomial Theorem: If n is a positive integer, then

n n X n! X  n  (x + y)n = xn−kyk = xn−kyk. (n − k)!k! k k=0 k=0 Remember that 0! = 1. Notice also that we have used the definition of combination.

Extended Permutation formula: The number of ways of partitioning n distinct objects into k distinct groups containing n1, n2, . . . nk objects, respectively is

k  n  n! X = , where ni = n. n1 n2 ··· nk n !n ! ··· n ! 1 2 k i=1

Multinomial Theorem: If n is a positive integer, then   n X n n1 n2 nk (x1 + x2 + ... + xk) = x1 x2 ··· xk . n1 n2 ··· nk

Pk Here, the sum is taken over all ni = 0, 1, . . . n such that i=1 ni = n.

1 Theorems to remember from calculus

The Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at x = a) and

lim f(x) = lim h(x) = L, x→a x→a then lim g(x) = L. x→a

The Intermediate Value Theorem: Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b) where f(a) 6= f(b). Then there exists a number c ∈ (a, b) such that f(c) = N.

The Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) ad an absolute value f(d) at some numbers c and d in [a, b].

Fermat’s Theorem: If f has a local maximum or minimum at c, and f 0(c) exists, then f 0(c) = 0.

Rolle’s Theorem: Let f be continuous on [a, b] and differentiable function on (a, b) with f(a) = f(b). Then there is a number c ∈ (a, b) such that f 0(c) = 0.

Mean Value Theorem: Let f be continuous on [a, b] and differentiable function on (a, b). Then there is a number c ∈ (a, b) such that

f(b) − f(a) f 0(c) = b − a or equivalently, f(b) − f(a) = f 0(c)(b − a).

The Fundamental Theorem of Calculus: Suppose that f is continuous on [a, b]. If we define R x 0 g(x) = a f(t)dt, then g (x) = f(x).

2 Important information on sequences and series

The Monotonic Sequence Theorem: Every bounded, monotonic sequence is convergent.

Geometric Series: The geometric series

∞ X arn−1 = a + ar + ar2 + ··· n=1 is convergent if |r| < 1 and it sum is

∞ X a arn−1 = . 1 − r n=1 Note that if |r| ≥ 1 the series is divergent.

P∞ Convergence Theorem: If the series n=1 an is convergent, then limn→∞ an = 0

Integral Test for Convergence: Suppose that f is a continuous, positive, decreasing function on P∞ [0, ∞) and let an = f(n). Then the series n=1 an is convergent if and only if the improper integral R ∞ 1 f(x)dx is convergent.

P∞ 1 P-Series: The p-series n=1 np is convergent if p > 1 and divergent if p ≤ 1.

Remainder Estimate for the Integral Test: Suppose that f(k) = ak, where f is a continuous, P positive, decreasing function on for x ≥ n and s = an converges. If Rn = s − sn where sn denotes the nth partial sum, then Z ∞ Z ∞ f(x)dx ≤ Rn ≤ f(x)dx. n+1 n

P P The Limit Comparison Test: Suppose that an and bn are series with positive terms. If a lim n = c n→∞ bn where c is finite and non-zero, then either both series converge or both diverge.

P∞ n−1 The Alternating Series Test: If the alternating series n=1(−1) bn satisfies bn+1 ≤ bn for all n and limn→∞ bn = 0, then the series is convergent.

Absolutely Convergent Theorem: If a series is absolutely convergent, then it is convergent.

3 The Ratio Test:

an+1 P∞ i) If limn→∞ = L < 1, then the series an is absolutely convergent. an n=1

an+1 P∞ ii) If limn→∞ = L > 1, then the series an is divergent. This also holds true if an n=1

an+1 limn→∞ = ∞. an

an+1 iii) If limn→∞ = 1, the test is inconclusive. an

The Root Test:

pn P∞ i) If limn→∞ |an| = L < 1, then the series n=1 an is absolutely convergent.

pn P∞ ii) If limn→∞ |an| = L > 1, then the series n=1 an is divergent. This also holds true if pn limn→∞ |an| = ∞.

pn iii) If limn→∞ |an| = 1, the test is inconclusive.

Taylor’s Formula Theorem: If f has a power series expansion at a, that is if

∞ X n f(x) = cn(x − a) , for |x − a| < R, n=0 then its coefficients are given by the formula

f (n)(a) c = . n n!

(n+1) Taylor’s Inequality: If f (x) ≤ M for |x − a| ≤ d, then the remainder Rn(x) of the Taylor series satisfies the inequality M |R (x)| ≤ |x − a|n+1 , for |x − a| ≤ d n (n + 1)!

4 Important series to remember

∞ X 1 xn = , for x ∈ (−1, 1) 1 − x n=0 ∞ X xn = ex, for x ∈ n! R n=0 ∞ X x2n+1 (−1)n = sin(x), for x ∈ (2n + 1)! R n=0 ∞ X x2n (−1)n = cos(x), for x ∈ (2n)! R n=0 ∞ X x2n+1 (−1)n = arctan(x), for x ∈ [−1, 1] 2n + 1 n=0 ∞ X xn (−1)n+1 = ln(1 + x), for x ∈ (−1, 1] n n=1 ∞ X 1 = 1 2n n=1 ∞ X 1 = 1 n(n + 1) n=1 n X n(n + 1) k = 2 k=1 n X n(n + 1)(2n + 1) k2 = 6 k=1 n 2 X n(n + 1) k3 = 2 k=1

5 Integrals to remember from calculus

Z ∞ √ e−x2 dx = π −∞ ∞ √ Z 2 π e−x dx = 0 2

Inequalities for Integrals

The Cauchy-Schwarz Inequality: Let f and g be square integrable functions. Then:

Z 2 Z  Z  f(x)g(x) dx ≤ f 2(x) dx g2(x) dx . D D D

Minkowski’s Inequality: If p > 1, then

1 1 1 Z  p Z  p Z  p |f(x) + g(x)|p dx ≤ |f(x)|p dx + |g(x)|p dx . D D D

1 1 H¨older’sInequality: If p, q > 1 such that p + q = 1, then

1 1 Z Z  p Z  q |f(x)g(x)| dx ≤ |f(x)|p dx |g(x)|q dx . D D D

Chebyshev’s Inequality: Let f and g be two increasing functions on R. Then for any real numbers a < b, Z b Z b  Z b  (b − a) f(x)g(x) dx ≥ f(x) dx g(x) dx a a a

6 Other Random Stuff

The Pigeonhole Principle: Given two natural numbers n and m with n > m, if n items are put into m pigeonholes, then at least one pigeonhole must contain more than one item.

Pick’s Theorem: Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon’s vertices are grid points, the area A of this polygon can be expressed in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon’s perimeter: b A = i + − 1 2

Integer Representation: Every natural number n can be written a n = 2km, where k ∈ {0, 1, 2,...} and m is an odd integer.

Triangle Geometry: The intersection of altitudes, the center of the circumscribed circle, and the centroid of the triangle are collinear.

Wallis Formula:  2 · 4 · 6 ··· 2n 2 1 lim · = π n→∞ 1 · 3 · 5 ··· (2n − 1) n

Stirling’s Approximation Formula: √ nn n! ∼ 2πn e

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