Statements Flashcards
Principle of Mathematical Induction
Let Sn be a statement about the positive integer n. Suppose that:
- S1 is true.
- Sk+1 is true whenever Sk is true.
Then Sn is true for all positive integers.
4 Cases of Integration by Partial Fractions.
- The denominator is a product of distinct linear factors.
- Den. is a product of linear factors, some repeated.
- Den. contains irreducible quadratic factors, none repeated. (complete the square + trig sub)
- Den. contains a repeated irreducible quadratic factor. (u-sub, trig sub).
Midpoint Rule
Use Riemann Sum. But Xi = (1/2)(x(i-1)+xi)
Xi = midpoint of [x(i-1_,xi)]
Trapezoidal Rule
Use Riemann Sum. Make trapezoids of Riemann Sums.
Error Bounds for Midpoint & Trapezoidal Rules
Suppose { |f’‘(x) <= K } for { a <= x <= b}. If Error(trap.) and Error(Mid.) are the errors in the Trapezoidal and Midpoint Rules, then:
| Error Mid | <= k(b-a)^3/24n^2
Error Trap | <= k(b-a)^3/12n^2
Definition of two types of Improper Integrals
Type 1: Infinite (double infinite) integrals, if limit exists then the integral is convergent. Otherwise, divergent if limit DNE.
Type 2: Vertical Asymptote Integrals. One-sided limits! Convergent if limit exists and divergent if limit DNE.
Monotonic Sequence Theorem
If a sequence is bounded and monotonic, then that sequence is convergent.
Definition of Bounded
A sequence {An} is bounded above if there is a number M such that {An <= M} for {all n >= 1}.
A sequence {An} is bounded below if there is a number M such that {An >= M} for all {n >= 1}.
A sequence that is both bounded above and below is called a bounded sequence.
Definition of Monotonic
A sequence {An} is called increasing if An < A(n+1) for all n>=1 (a1 < a2 < a3).
A sequence {An} is called decreasing if An > A(n+1) for all
n >= 1. (a1 > a2 > a3).
A sequence is monotonic if it is increasing or decreasing.
Limit Divergence Test
If the limit of a series is not 0, then it is not convergent.
The Integral Test
If f is a continuous, positive, decreasing function on [1, inf) and let a(n) = f(n). Then the series {an} is convergent IF AND ONLY IF its improper integral is convergent.
Aka
If the improper integral is convergent, the series is convergent.
If the improper integral is divergent, the series is also divergent.
Remainder Estimate for the Integral Test
Suppose f(k) = ak, where f is a continuous, positive, decreasing function for x >= n and the series ak is convergent.
If Rem n = S - Sn, then
Rem < = integral (n,inf) f(x) dx
Comparison Test
Suppose {An} and {Bn} are series with POSITIVE terms. Then:
- If {Bn} is convergent and An <= Bn for all n, then {An} is also convergent.
Likewise, 2. If {Bn} is divergent and An >= Bn for all n, then {Bn} is also divergent.
Limit Comparison Test
Suppose that {An} and {Bn} are series with POSITIVE TERMS.
If the infinite limit of An/Bn = C where C is a finite number and C IS STRICTLY GREATER THAN 0…
then the series An and Bn both diverge or both converge.
Alternating Series Test
If an alternating series satisfies:
- b(n+1) <= bn for all n and
- Infinite limit of bn = 0.
… then the series is convergent.
