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--- math104-s21:s:franceskim [2021/05/12 06:11]
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+++ math104-s21:s:franceskim [2026/02/21 14:41] (current)
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 ii) lim g(x) = +inf as x -> a
-**Lecture 20**
+**Lecture 21**
 A) Smooth functions:
@@ Line 491: / Line 491: @@
 Derivatives exist to all order.
-.
+B) Taylor Theorem:
+f: [a, b] -> R is a function s.t. f^(n-1) (x) exists and is continuous on [a, b]. f^(n) (x) exists on (a, b).
+Then, for any α, β ∈ [a, b], we have:
+f(β) = f(α) + f'(α)(β-α) + f''(α)/2!(β-α)^2 + ... + f^(n-1)(α)/(n-1)!(β-α)^(n-1) + R_n(α, β)
+R_n(α, β) = 0 if α = β, R_n(α, β) = f^n(r)/n!(β-α)^n if α ≠ β for some r ∈ (α, β)
+**Lecture 22**
+A) Power series:
+Series of the form ∑ C_n(x-x_0)^n from n = 0 to n = +inf
+B) Radius of Convergence R:
+R = sup {r >= 0, s.t. if |x-x_0| <= r, the series converges}
+C) If R = 1/a, where a = limsup|C_n|^(1/n) as n -> +inf:
+If |x-x_0| < R, the series converges.
+If |x-x_0| > R, the series diverges.
+. Prove the diverging case of (C).
+D) Real Analytic function f:
+f: (a, b) -> R is smooth, for all x_0 ∈ (a, b), f(x) = ∑ C_n(x-x_0)^n for x in some neighborhood of x_0.
+. Regarding the definition of real analytic function, what does it mean for "f(x) = ∑ C_n(x-x_0)^n for x in some neighborhood of x_0"?
+E) Length of a function:
+∫sqrt(1 + (f'(x))^2)dx
+F) U(P, f) = ∑ M_i*Δx_i from i = 1 to i = n
+L(P, f) = ∑ m_i*Δx_i from i = 1 to i = n
+M_i = sup{f(x)|x ∈ [x_(i-1), x_i]}
+m_i = inf{f(x)|x ∈ [x_(i-1), x_i]}
+Δx_i = x_i - x_(i-1)
+G) U(f) = inf U(P, f)
+L(f) = sup L(P, f)
+H) f is integrable if U(f) = L(f)
+I) Generalization of Riemann-Stieltjes integrable:
+Let α: [a, b] -> R be a monotone increasing function, define partition P = {a = x_0 <= x_1 <= ... <= x_n = b}, define Δα_i = α(x_i) - α(x_(i-1))
+The remaining definitions are similar as in parts (F) and (G), except Δx_i -> Δα_i, and U(P, α) = L(P, α) implies that f is Riemann-Stieltjes integrable w.r.t. α.
+**Lecture 23**
+A) If a partition Q is a refinement of partition P on [a, b], then L_P <= L_Q <= U_Q <= U_P.
+B) L(f, α) <= U(f, α)
+C) f is integrable w.r.t. α iff for all ε > 0, there exists P partition s.t. U_P - L_P < ε
+.
 General Note:

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