Real Analysis and Calculus Notes
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Sequence
A nonempty set of real numbers that is bounded above has a least upper
bound, i.e. supremums of bounded sets are real numbers.
Every sequence contains a monotonic subsequence.
Suppose that \(\left\{x_{n}\right\}\) is a monotonic sequence. Then,
\(\left\{x_{n}\right\}\) is convergent if and only if
\(\left\{x_{n}\right\}\) is bounded.
Every bounded sequence contains a convergent subsequence.
A sequence \(\left\{x_{n}\right\}\) is convergent iff for each
\(\varepsilon>0\) there exists an integer \(N\) with the property that
Converge Tests
Null test
If the terms of the series \(\sum_{k=1}^{\infty} a_{k}\) do not converge
to zero, then the series diverges.
Comparison test
Given two series \(\sum_{k=1}^{\infty} a_{k}\) and
\(\sum_{k=1}^{\infty} b_{k}\) such that \(0 \leq a_{k} \leq b_{k}\) for all
\(k\).
1. If the larger series converges, then so does the smaller series.
2. If the smaller series diverges, then so does the larger series.
Ratio test
If terms of the series \(\sum_{k=1}^{\infty} a_{k}\) are all positive and
the ratios $$\lim {k \rightarrow \infty} \frac{a{k+1}}{a_{k}}<1$$ then
the series is convergent.
Root test
If terms of the series \(\sum_{k=1}^{\infty} a_{k}\) are all nonnegative
and the roots $$\lim {k \rightarrow \infty} \sqrt[k]{a{k}}<1$$ then
the series is convergent.
Integral test
Let \(f\) be a nonnegative decreasing function on \([1, \infty) .\) Then
\[\lim _{X \rightarrow \infty} \int_{1}^{X} f(x) d x$$ converges if and only if the series $\sum_{k=1}^{\infty} f(k)$ converges. #### Proof since $f$ is decreasing we have $$\int_{k}^{k+1} f(x) d x \leq f(k) \leq \int_{k-1}^{k} f(x) d x$$ Thus $$\int_{1}^{n+1} f(x) d x \leq \sum_{k=1}^{n} f(k) \leq f(1)+\int_{1}^{n} f(x) d x\]The series converges if and only if the partial sums are bounded.
Alternating Series test
The series $$\sum_{k=1}{\infty}(-1){k-1} a_{k}$$ where the terms
alternate in sign, converges if the sequence \(\left\{a_{k}\right\}\)
decreases monotonically to zero.
Power Series
A power series
\[f(x) \mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\sffamily def}}}{=}}\sum _{n=0}^\infty a_n x^n, x\in S \]where \(S\) will make sense.
Radius of Convergence
Given a power series \(\sum_{n=0}^{\infty} a_{n} x^{n},\) either it
converges absolutely for all \(x \in \mathbb{R},\) or there exists
\(R \in[0, \infty)\) such that
(1) it converges absolutely when \(|x|<R\)
(2) it diverges when \(|x|>R\).
Remark
We can restate the theorem as $$(-R, R) \subseteq S \subseteq[-R, R]$$
and the power series converges absolutely in \((-R, R) .\) In particular
we see that \(S\) is always an interval.
Convergence Test (of Power Series)
Ratio test (of Power Series)
Consider the power series $$\sum_{n=0}^{\infty} a_{n} x^{n}$$ Suppose
that
Then $$R=\left{\begin{array}{ll}
0 & \text { if } \quad \ell=\infty \
\frac{1}{\ell} & \text { if } \quad \ell \in \mathbb{R} \backslash{0} \
\infty & \text { if } \quad \ell=0
\end{array}\right.$$
Root test (of Power Series)
Consider the power series $$\sum_{n=0}^{\infty} a_{n} x^{n}$$ Suppose
that
Then $$R=\left{\begin{array}{ll}
0 & \text { if } \quad \ell=\infty \
\frac{1}{\ell} & \text { if } \quad \ell \in \mathbb{R} \backslash{0} \
\infty & \text { if } \quad \ell=0
\end{array}\right.$$
Maclaurin and Taylor Series
Definition
If the function \(f\) has a power series representation on the interval
\((c-R, c+R),\) then the power series $$\begin{aligned}
f(x) &=\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n !}(x-c)^{n} \
&=\frac{f(c)}{0 !}+\frac{f^{\prime}(c)(x-c)}{1 !}+\frac{f^{\prime \prime}(c)(x-c)^{2}}{2 !}+\frac{f^{\prime \prime \prime}(c)(x-c)^{3}}{3 !}+\cdots
\end{aligned}$$ is called the Taylor Series of the function \(f\) about
\(c\). In the particular case that \(c=0,\) then Taylor series of \(f\) is
usually called the Maclaurin series of \(f:\) $$\begin{aligned}
f(x) &=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n !} x^{n} \
&=\frac{f(0)}{0 !}+\frac{f^{\prime}(0) x}{1 !}+\frac{f^{\prime \prime}(0) x^{2}}{2 !}+\frac{f^{\prime \prime \prime}(0) x^{3}}{3 !}+\cdots
\end{aligned}$$\
Things must be Memorized
1. For any \(x \in \mathbb{R}\)
\[e^{x}=1+x+\frac{x^{2}}{2}+\cdots+\frac{x^{n}}{n !}+\cdots=\sum_{n=0}^{\infty} x^{n} / n ! \]- For any \(x \in \mathbb{R}\)\[\sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\cdots+(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} x^{2 n+1} /(2 n+1) ! \]
- For any \(x \in \mathbb{R}\)\[\cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{4 !}+\cdots+(-1)^{n} \frac{x^{2 n}}{(2 n) !}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} x^{2 n} /(2 n) ! \]
- The Binomial Theorem: for any \(\alpha \in \mathbb{R}\) and \(x\) such
that \(|x|<1\) $$\begin{aligned}
(1+x)^{\alpha} &=1+\alpha x+\frac{\alpha(\alpha-1)}{2} x^{2}+\cdots+\frac{\alpha(\alpha-1) \ldots(\alpha-n+1)}{n !} x^{n}+\cdots \
&=\sum_{n \geq 0} \frac{\alpha(\alpha-1) \cdots(\alpha-n+1)}{n !} x^{n}
\end{aligned}$$ 5. From \(4 .\) we have, for any \(x\) such that \(|x|<1\),
6. For any \(x\) such that \(|x|<1\)
\[\log (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots+(-1)^{n+1} \frac{x^{n}}{n}+\cdots=\sum_{n \geq 1}(-1)^{n+1} \frac{x^{n}}{n} \]Things Must be Mentioned about Series
\(\sum a_n\) converges\(\overset{???}\iff \sum a_n^3\) converges
\(\not \Rightarrow\)
\(\sum a_n\) converges does not imply \(\sum a_n^3\) converges in
general for non positive \(a_n\). For an \(m\), write as \(m=3n+k\) where
\(0\leq k<3\) and define \(a_{3n+k}= \frac{b_k}{(n+1)^{1/3}}\) where \(b_0=2\)
and \(b_1,b_2=-1\). Then \(a^3_n\) in general looks like
which has partial sums \(S_{3n}=6\sum _{i=1}^{\infty} 1/i\) diverging.
\(\not \Leftarrow\)
\[\sum \frac{1}{k^3}\quad \textit{converges} \not \Rightarrow \sum \frac{1}{k}\quad \textit{converges} \]Basic Integration Methods
Some Other Things
Hyperbolic Function
Definition
\[\sinh x :=\frac{e^x-e^{-x}}{2}\qquad \cosh x:=\frac{e^x+e^{-x}}{2}\]Identities of Trigonometric Functions
Pythagorean Identities
\[\begin{aligned} \sin^2 \theta + \cos^2 \theta=& 1\\ \tan^2 + 1=&\sec^2 \theta \\ 1+ \cot ^2=& \csc^2 \theta \end{aligned}\]Recite these Equations
For integrate is the inverse of the derivative, we just have to recite
some basic rules about the derivative. Equation below request to be
recited. $$\begin{aligned}
(\arcsin x)'&=\frac{1}{\sqrt{1-x^2}}\
(\arccos x)'&=-\frac{1}{\sqrt{1-x^2}}\
(\arctan x)'&=\frac{1}{1+x^2}\
(\tan x)'&=\sec^2 x\
(\cot x)'&=-\csc^2 x\
(\sinh x)'&= \cosh x\
(\cosh x)'&= \sinh x\
\end{aligned}$$
Some Classic Integrations
The following equation have some interesting conclusion.
\[\begin{aligned} \int \sec x\mathop{}\!\mathrm{d}x &=\ln \left|\frac{\cos \frac{x}{2} + \sin \frac{x}{2}}{\cos \frac{x}{2} -\sin \frac{x}{2}}\right|+C\\ &= \ln \left| \frac{1+\sin x}{\cos x}\right|+C \\ &= \ln \left| \sec x+ \tan x \right| + C \end{aligned}\]Riemannn Integrable
Continuous[1] \(\Rightarrow\) Riemann Integrable \(\Rightarrow\) Boundness
\(\Uparrow\)
Monotone
Some Particular Functions
Weierstrass function
the Weierstrass function is an example of a real-valued function that is
continuous everywhere but differentiable nowhere. In Weierstrass's
original paper, the function was defined as a Fourier series:
Volterra's function
The function is defined by making use of the Smith Volterra Cantor set
and "copies" of the function defined by \(f(x)=x^{2}\sin(1/x)\) for
\(x\neq 0\) and \(f(0)=0\).
-
\(V\) is differentiable everywhere
-
The derivative \(V'\) is bounded everywhere
-
The derivative is not Riemann-integrable.
On closed set ↩︎
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