calc 2

What is theorem 2: The Sandwich Theorem for Sequences

• Let {a"}, {b"}, and {C”} be sequences of real numbers. If an

= Cn holds for all n beyond some index N, and if lim(n→inf) an = lim(n→inf) cn = L, then lim(n→inf) bn= L also

What is theorem 3: The Continuous Function Theorem for Sequences

• Let {an) be a sequence of real numbers. If an → L and if f is a function that is continuous at L and defined at all an, then f(a) → f(L)

What is theorem 1: rules

Let {a} and {b} be sequences of real numbers, and let A and B be real numbers. The following rules hold if lim(n→inf) an= A and lim(n→inf) bn= B.

What is theorem 4: l’hopitals rule

• Suppose that f(x) is a function defined for all x > n0 and that {a" ) is a sequence of real numbers such that a, = f(n) for n ≥ n0.

• Then lim(n→inf) an = L whenever lim(x→inf) f(x) = L

• Always l’hopitals rule

What is theorem 6: The Monotonic Sequence

• If a sequence {an } is both bounded and monotonic, then the sequence converges.

What is geometric series

• If Irl ‹ 1, the geometric series a + ar^(2) +…+ ar^(n-1)+ ... converges to 1 to inf a/(1-r), lrl < 1

• If Irl ≥ 1, the series diverges.

What is theorem 9: Integral test

• Let {a) be a sequence of positive terms. Suppose that an= f(n), where f is a continuous, positive. decreasing function of x for all x ≥ N (N a positive integer). Then the series both diverge.

Let an and bn be two series with 0 ≤ an ≤ bn for all n.

1. If bn converges, then an also converges.

2. If a diverges, then bn also diverges.

What is theorem 11: limit comparison test

Suppose that an > 0 and bn > 0 for all n ≥ N (N an integer).

1.(a/b) If lim(N→inf) = c and c > O, then an/bn both converge or both diverge.

2.(a/b) If lim(n→inf)= 0 and bn, converges then an converges

3.(a/b) if lim(n→inf)=0 bn, diverges then an diverges

What is theorem 12: absolute convergence test

• If lim(n→inf) lanl converges so does lim(n→inf) an

What is theorem 13: ratio test

• If lim(n→inf) la(n+1)/anl =p

• Then (a) the series converges absolutely if p ‹ 1, (b) the series diverges if p > 1 or p is infinite, and (c) the test is inconclusive if p = 1.

What is theorem 14: root test

• If lim(n→inf) sqrt(lanl) =p

• Then (a) the series converges absolutely if p < 1, (b) the series diverges if p > 1 or p is infinite, and (c) the test is inconclusive if p = 1.

What is theorem 15: alternating series test

If lim(n→inf) (-1)^(n+1)un = u1 - u2 + u3 - u4 +…

converges if the following conditions are satisfied:

1. The u,'s are all positive.

2. The u,'s are eventually nonincreasing: U,

2. The un ≥ un+1 for all n ≥ N, for some integer N.

3. Un → О.

What is theorem 18: The Convergence Theorem for Power Series

If the power series:

• anx^(n)= a0 + a1x + a2x^(2) + ... converges at x = c ≠0, then it converges

• absolutely for all x with |x| < Icl

• If the series diverges at x = d, then it diverges for all x with |xl > Idl

What is theorem 19: Series Multiplication for Power Series

• If A(x)=an x^(n) and B(x)=bn x^(n) converge for lxl<R

• Cn=a0bn + a1bn-1 …+…+ akbn-k

• Then CnXn converge is absolutely to A(x) B(x) for lxl<R

What theorem 20

• anx^n converges absolutely for |x| < R, and f is a

continuous function

• Then anf(x))^n converges absolutely on the set of points x that satisfy |f(x)| < R.

What is theorem 21: term by term differentiation

• Cn(x - a)^n has radius of convergence R > 0, it defines a function f(x) = Cn(x-a)^n on the interval a - R < x < a + R.

• This function f has derivatives of all orders inside the interval, and we obtain the derivatives by differentiating the original series term by term:

• f’(x) nCn(x-a)^(n-1) and f”(x)=n(n-1)Cn(x-a)^(n-2)

• and so on. Each of these derived series converges at every point of the interval

  a - R< x < at R.

What is theorem 22: term by term integration

• If f(x)=Cn(x-a)^n

• converges for a - R < x < a + R (where R > 0). Then (x-a)^n/n+1

• converges for a - R < x < a + Rand

• f(x) dx= (x-a)^n/n+1 +C for a - R < x < a + R.