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| title | showthedate | math |
|---|---|---|
| Real Analysis Quick Sheet | false | true |
Fact: \forall a,b, \in \mathbb{R}, \sqrt{ab} \le \frac{1}{2}(a + b).
Bernoulli's Inequality: If x > -1, then (1 + x)^n \ge 1 + nx, \forall n \in \mathbb{N}.
Triangle Inequality: If a,b \in \mathbb{R}, then |a + b| \le |a| + |b|.
Epsilon Neighborhood Definition:
Let a \in \mathbb{R} and \epsilon > 0. The $\epsilon$-neighborhood of a is the set V_\epsilon(a) = \{ x \in \mathbb{R}, |x - a| < \epsilon\}.
Archimedean Property: If x \in \mathbb{R}, then \exists n_x \in \mathbb{N} such that x \le n_x.
Convergence Definition:
X = (x_n) converges to x \in \mathbb{R} if \forall \epsilon > 0, \exists k \in \mathbb{N} such that \forall n \ge k, |x_n - x| < \epsilon.
Fact: A sequence in \mathbb{R} can have at most one limit.
Theorem 3.1.9:
If (a_n) is a sequence in \mathbb{R} such that a_n \rightarrow 0, and for some constant c > 0, m \in \mathbb{N},
|x_n - x| \le ca_n, \forall n \ge m, then x_n \rightarrow x.
Theorem 3.2.7:
(a) If x_n \rightarrow x, then |x_n| \rightarrow |x|.
(b) If x_n \rightarrow x and x_n \ge 0, then \sqrt{x_n} \rightarrow \sqrt{x}.
Ratio Test:
Let \{x_n\} \subseteq \mathbb{R}^+ such that L = \lim{(\frac{x_{n + 1}}{x_n})}. If L < 1, then x_n \rightarrow 0.
Theorem 3.2.2: Every convergent sequence is bounded. (The converse is not necessarily true)
Squeeze Theorem:
If x_n \rightarrow x, y_n \rightarrow y, and z_n \rightarrow x such that x_n \le y_n \le z_n, \forall n \in \mathbb{N} then y = x.
Monotone Convergence Theorem
Let X = (x_n) be a subsequence in \mathbb{R}.
(a) If X is monotonically increasing and bounded above then \lim{x_n} = sup\{x_n : n \in \mathbb{N}\}
(b) If X is monotonically decreasing and bounded below then \lim{x_n} = inf\{x_n : n \in \mathbb{N}\}
Useful Fact: If \lim{x_n} = a \in \mathbb{R}, then \lim{x_{n + 1}} = a.
Interesting Application of MCT:
Let s_1 > 0 be arbitrary, and define s_{n + 1} = \frac{1}{2}(s_n + \frac{a}{s_n}). We know s_n \rightarrow \sqrt{a}.
Euler's Number: Consider the sequence e_n = (1 + \frac{1}{n})^n. We know e_n \rightarrow e.
Theorem 3.4.2: If X = (x_n) \subseteq \mathbb{R} converges to x, then every subsequence converges to x.
Corollary: If X = (x_n) \subseteq \mathbb{R} has a subsequence that diverges then X diverges.
Monotone Sequence Theorem: If X = (x_n) \subseteq \mathbb{R}, then it contains a monotone subsequence.
Bolzano-Weierstrass Theorem: Every bounded sequence in \mathbb{R} has a convergent subsequence.
Theorem 3.4.12: A bounded (x_n) \in \mathbb{R} is convergent iff liminf(x_n) = limsup(x_n).
Cauchy Criteria for Convergence: Let X = (x_n) \subseteq \mathbb{R}. We say that X is cauchy if \forall \epsilon > 0, \exists N \in \mathbb{N} such that \forall n,m \ge N, |x_n - x_m| < \epsilon. We know that a sequence converges iff it is cauchy.
P-Series: Series of the form s_n = \sum_{i = 0}^n{\frac{1}{n^p}} is convergent for p > 1.
Geometric Series:
Series of the form s_n = \sum_{i = 0}^n{ar^i} has the following partial sum a\frac{1 - r^n}{1 - r} and converges to \frac{a}{1 - r} if |r| < 1.
Comparison Test:
Let (x_n), (y_n) \subseteq \mathbb{R}. Suppose that for some k \in \mathbb{N}, 0 \le x_n \le y_n, for n \ge k.
(a) If \sum{y_n} < \infty, then \sum{x_n} < \infty.
(b) If \sum{x_n} = \infty, then \sum{y_n} = \infty.
Limit Comparison Test:
Let (x_n), (y_n) be strictly positive sequence of real numbers. Suppose r = \lim{\frac{x_n}{y_n}}.
(a) If r \ne 0, then \sum{x_n} < \infty \iff \sum{y_n} < \infty.
(b) If (r = 0 and \sum{y_n} < \infty), then \sum{x_n} < \infty.