Newton's method

Used to approximate the root of a function. The update rule:

\[ \begin{aligned} x_{n+1} = x_n - \frac{f(x)}{f'(x)} \end{aligned} \]

See more details at PennCalcWiki.

Power Series

Given a sequence \(a_n\), the power series assoiciated with \(a_n\) is the infinite sum

\[ \begin{aligned} f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \cdots \end{aligned} \]

Interval and radius of convergence

Given a power series, there exists some \(0 \le R \le \infty\) such that

• the series converges absolutely if \(\lvert x \rvert < R\)
• the series diverges if \(\lvert x \rvert > R\)
• might converge or diverge when \(x = R\) or \(x = -R\)

The interval of convergence is the interval \((-R, R)\), possibly with the endpoints included (they need to be individually checked in general). The radius of convergence is half the length of the interval of convergence.

To find the interval of convergence, use ratio test or root test to find the interval where the series converges absolutely and then check the endpoints of the interval. For example,

\[ \begin{aligned} \rho &= \lim_{n \rightarrow \infty} \frac{\lvert a_{n+1}x^{n+1} \rvert}{\lvert a_n x^n \rvert} = \lim_{n \rightarrow \infty} \frac{\lvert a_{n+1} \rvert}{\lvert a_n \rvert} \lvert x \rvert < 1 \\ \lvert x \rvert &< \lim_{n \rightarrow \infty} \frac{\lvert a_n \rvert}{\lvert a_{n+1} \rvert} = R \hskip2em \text{(radius of convergence)} \end{aligned} \]

Shifted power series

\[ \begin{aligned} f(x) &= a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots \\ \lvert x - c\rvert &< \lim_{n \rightarrow \infty} \frac{\lvert a_n \rvert}{\lvert a_{n+1} \rvert} = R \end{aligned} \]

Interval of convergence: \((c - R, c + R)\). Such a power series is said to be centered at \(c\), since the interval of convergence for the series will be centered at \(c\).

Taylor series convergence

For what values of \(x\) does a function's Taylor series converge ?

The series:

\[ \begin{aligned} \sum_{n=0}^\infty a_n x^n &= \sum_{n=0}^\infty \frac{1}{n!} f^{(n)}(0) \, x^n \\ \end{aligned} \]

converges absolutely for \(\lvert x \rvert < R \), where

\[ \begin{aligned} R = \lim_{n \rightarrow \infty} \frac{\lvert a_n \rvert}{\lvert a_{n+1} \rvert} = \lim_{n \rightarrow \infty} \frac{\lvert f^{(n)}(0) \rvert }{\lvert f^{(n+1)}(0) \rvert} \cdot \frac{(n+1)!}{n!} = \lim_{n \rightarrow \infty} (n+1) \cdot \frac{\lvert f^{(n)}(0) \rvert }{\lvert f^{(n+1)}(0) \rvert} \\ \end{aligned} \]

Within the interval of convergence, differentiation and integration of a power series are nice, in that they can be done term by term.

When a Taylor series converges, it does not always converge to the function. But when it does, the function is referred to as real-analytic. That is, a function \(f\) is real-analytic at \(a\) if the Taylor series for \(f\) converges to \(f\) near \(a\).

Error defined

Given a convergent series,

\[ \begin{aligned} E_k &= s - s_k \\ &= \sum_{n=0}^\infty a_n - \sum_{n=0}^k a_n \\ &= \sum_{n=k+1}^\infty a_n \\ \end{aligned} \]

Alternating series error bound

\[ \begin{aligned} \lvert E_k \rvert \le \lvert a_{k+1} \rvert \\ \end{aligned} \]

Integral test for error bounds

\[ \begin{aligned} \int_{m+1}^\infty f(x)\,dx < \sum_{n=m+1}^\infty f(n) = E_m < \int_{m}^\infty f(x)\,dx \\ \end{aligned} \]

Taylor approximations

\[ \begin{aligned} f(x) &\approx \sum_{n=0}^N \frac{f^{(n)}(0)}{n!}\,x^n \\ f(x) &= \sum_{n=0}^N \frac{f^{(n)}(0)}{n!}\,x^n + E_N(x) \\ \end{aligned} \]

Weak bound

\[ \begin{aligned} E_N(x) \le C\,x^{N+1} = O(x^{N+1}) \end{aligned} \]

for some constant \(C\) and \(x\) sufficiently close to 0.

Taylor Remainder Theorem

\[ \begin{aligned} E_N(x) = \frac{f^{(n+1)}(t)}{(N+1)!}x^{N+1} \end{aligned} \]

for \(\lvert t \rvert \le \lvert x \rvert \), which is not very useful since there is no way to determine \(t\).

Taylor error bound

\[ \begin{aligned} \lvert E_N(x) \rvert \le \frac{max(\lvert f^{(n+1)}(t) \rvert)}{(N+1)!}\lvert x^{N+1} \rvert \end{aligned} \]

for \(\lvert t \rvert \le \lvert x \rvert \).

Partial sum

\(\displaystyle \sum_{n=1}^T a_n\) is called the \(T\)th partial sum of the series \(\displaystyle \sum_{n=1}^\infty a_n\). A series converges if the sequence of partial sums converges.

\(n\)th term test

If \(\displaystyle \lim_{n \rightarrow \infty} a_n \ne 0\), then the series \(\displaystyle \sum_{n=0}^\infty a_n\) diverges. Can be used to test for divergence (but not convergence). ie The converse is not always true, but the contra-positive is.

Integral test

\[ \begin{aligned} \sum_{n=m}^\infty f(n) \hskip1em \text{converges} \hskip2em \Leftrightarrow \hskip2em \int_{m}^\infty f(x) \, dx \hskip1em \text{converges} \\ \end{aligned} \]

Condition: \(f(x)\) is a positive decreasing function.

p-series test

\[ \begin{aligned} \sum_{n=1}^\infty \frac{1}{n^p} \hskip1em \text{converges if and only if } p > 1 \\ \end{aligned} \]

Comparison test

Let \(a_n\) and \(b_n\) be positive sequences such that \(b_n > a_n\) for all \(n\). It follows that if \(\sum b_n\) converges, then so does \(\sum a_n\), and if \(\sum a_n\) diverges, then so does \(\sum b_n\).

Limit test

Let \(a_n\) and \(b_n\) be positive series. If \(\displaystyle \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = L \) and \(0 < L < \infty\), then the series \(\sum a_n\) and \(\sum b_n\) either both converge or both diverge.

Root test

Given a series \( \sum a_n\) with all \(a_n > 0\), let \(\displaystyle \rho = \lim_{n \rightarrow \infty} \sqrt[n]{a_n} \). If \(\rho\) < 1, then \(\sum a_n\) converges. If \(\rho\) > 1, then \(\sum a_n\) diverges. If \(\rho\) = 1, then the test is inconclusive. Based on the idea of "eventual geometric series".

Ratio test

Given a series \( \sum a_n\) with all \(a_n > 0\), let \(\displaystyle \rho = \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} \). If \(\rho\) < 1, then \(\sum a_n\) converges. If \(\rho\) > 1, then \(\sum a_n\) diverges. If \(\rho\) = 1, then the test is inconclusive. Work best when exponential or factorial factor is involved - not so with ratio of polynomials.

Summary of methods for a positive series

Given a series \(\sum a_n\), where \(a_n > 0\) for all \(n\):

1. Do the terms go to 0 ? If \(\displaystyle \lim_{n \rightarrow \infty} a_n \ne 0 \), then by the nth term test, the series diverges. (If \(\displaystyle \lim_{n \rightarrow \infty} a_n = 0 \), then the test is inconclusive).
2. Does \(a_n\) involve exponential functions like \(c^n\) where \(c\) is constant ? Does \(a_n\) involve factorial ? Then the ratio test should be used.
3. Is \(a_n\) of the form \((b_n)^n\)for some sequence \(b_n\) ? Then use the root test.
4. Does ignoring lower order terms make \(a_n\) look like a familiar series (e.g. p-series or geometric series) ? Then use the comparison test or the limit test.
5. Does the sequence \(a_n\) look easy to integrate ? Then use the integral test.

Alternating Series test

An alternating series \( \sum (-1)^n b_n\) with decreasing terms converges if and only if \(\displaystyle \lim_{n \rightarrow \infty} b_n = 0 \). (Note when to examine the asymptotic value of the \(n\)th term vs. examining the p-value of an integral.)

Conditional vs Absolute Convergence

\(\displaystyle \sum a_n\) converges absolutely if \(\displaystyle \sum \lvert a_n \rvert \) converges. \(\displaystyle \sum a_n\) converges conditionally if \(\displaystyle \sum \lvert a_n \rvert \) diverges but \(\displaystyle \sum a_n \) converges. In other words, a series is conditionally convergent if it is convergent but not absolutely convergent.

Absolute Convergence test

If a series converges absolutely, then it always converges. (But this is not necessarily true if it converges conditionally.)

Interesting notes

• \(\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{n} = 1\)

  \[ \begin{aligned} y &= \lim_{n \rightarrow \infty} \sqrt[n]{n} = \lim_{n \rightarrow \infty} n^\frac{1}{n} \\ ln(y) &= \lim_{n \rightarrow \infty} \frac{1}{n} ln(n) = 0 \\ \therefore y &= 1 \\ \end{aligned} \]

• \(\displaystyle \lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n = e = \sum_{n=0}^\infty \frac{1}{n!} \)
• \(\displaystyle \lim_{n \rightarrow \infty} (1 + \frac{a}{n})^n = e^a = \sum_{n=0}^\infty \frac{1}{n!} a^n \)
• \(\displaystyle 0 < ln(2) < 1 \)

Product Rule

\[ \begin{aligned} \Delta (ab)_n &= a_n \Delta b_n + b_n \Delta a_n + \Delta a_n \Delta b_n \\ \triangledown (ab)_n &= a_n \triangledown b_n + b_n \triangledown a_n - \triangledown a_n \triangledown b_n \\ \\ \Delta (uv) &= u \Delta v + Ev \Delta u \\ \end{aligned} \]

If \(a\) is a polynomial of degree \(p\), then

\[ \begin{aligned} \Delta^{p+1} a &= (0) \\ \end{aligned} \]

Falling Power Rule

\[ \begin{aligned} n^{\underline k} &= n (n-1) \cdots (n-k+1), \hskip4em n^\underline{0} = 1 \\ \Delta n^{\underline k} &= k \, n ^{\underline{k-1}}\\ n^{- \underline k} &= \frac{1}{(n+1) (n+2) \cdots (n+k)} \\ \end{aligned} \]

Sequence operators

\[ \begin{aligned} (Ia)_n &= a_n \\ (Ea)_n &= a_{n+1} \\ (\Delta a)_n &= a_{n+1} - a_n, \hskip4em \Delta = E - I \\ (\triangledown a)_n &= a_n - a_{n-1}, \hskip4em \triangledown = I - E^{-1} \\ (\triangledown \Delta a)_n &= a_{n+1} - 2a_n + a_{n-1}, \hskip1em \text{second central difference} \\ \triangledown \Delta \, a &= \Delta \triangledown a \\ \end{aligned} \]

Higher Derivatives

\[ \begin{aligned} \Delta^k &= (E - I)^k = \sum_{i=0}^k (-1)^{k-i} {k \choose i} E^i \\ \end{aligned} \]

Anti-difference

\[ \begin{aligned} \Delta^{-1} &= (E - I)^{-1} = -(I - E)^{-1} = -(I + E + E^2 + E^3 + \cdots) + C \\ \end{aligned} \]

so long as the sequence \(a_n\) is eventually 0

Discrete Fundamental Theorem of Integral Calculus

\[ \begin{aligned} \sum_{n=A}^B \Delta u &= u \, \bigg \rvert_{n=A}^{B+1} \\ \sum_{n=A}^B \triangledown u &= u \, \bigg \rvert_{n=A-1}^{B} \\ \end{aligned} \]

Alternatively,

\[ \begin{aligned} \sum_{n=A}^B u &= \Delta^{-1} u \, \bigg \rvert_{n=A}^{B+1} \\ \sum_{n=A}^B u &= \triangledown^{-1} u \, \bigg \rvert_{n=A-1}^{B} \\ \end{aligned} \]

Integration by parts

\[ \begin{aligned} \Delta (uv) &= u \Delta v + Ev \Delta u \\ \sum_{n=A}^B u \Delta v &= uv \, \bigg \rvert_{n=A}^{B+1} - \sum_{n=A}^B Ev \Delta u \\ \end{aligned} \]

Differential equations

Linear first-order recurrence relation:

\[ \begin{aligned} u_{n+1} &= \lambda u_n \\ E \, u &= \lambda u \\ (E - \lambda \, I) \, u &= 0 \\ \implies u_n &= u_0 \, \lambda^n \\ \end{aligned} \]

Linear first-order difference equation:

\[ \begin{aligned} \Delta u &= \lambda u \\ (E - I) \, u &= \lambda u \\ (E - (\lambda + 1) \, I) \, u &= 0 \\ \implies u_n &= u_0 \, (\lambda + 1)^n \\ \end{aligned} \]

Numerical ODE's

Many differential equations of the form \(\displaystyle \frac{dx}{dt} = f(x,t) \) cannot be solved exactly. Hence the use of numerical ODE's to approximate a solution.

Euler's method

Uses a difference equation to approximate the solution of an initial value problem. Essentially it's Taylor series 1st order approximation for the recursion relation. Given \(\displaystyle \frac{dx}{dt} = f(x,t) \) , and initial value \( x_0 = x(t_0) \) , Euler's method approximates \( x(t_*) \) for some \(t_* > t_0 \).

\[ \begin{aligned} x_{n+1} = x_n + h\,f(x_n, t_n) \\ t_{n+1} = t_n + h \\ h = \frac{t_* - t_0}{N} \\ \end{aligned} \]

where N is the number of intervals.

Numerical Integration

How to approximate the definite integral \(\displaystyle \int_a^b f(x)\,dx\) via finite sums which approximates the area under the curve ? Common techniques: Riemann sums, trapezoid rules and Simpson's rule.

Left Riemann Sum

\[ \begin{aligned} \int_a^b f(x)\,dx \approx \sum_{n=0}^{N-1} f_n \cdot (\Delta x)_n = \sum_{n=0}^{N-1} f_n \cdot (x_{n+1} - x_n) \\ \end{aligned} \]

Right Riemann Sum

\[ \begin{aligned} \int_a^b f(x)\,dx \approx \sum_{n=1}^{N} f_n \cdot (\triangledown x)_n = \sum_{n=1}^{N} f_n \cdot (x_n - x_{n-1}) \\ \end{aligned} \]

Trapezoid Rule

An improvement over the left and right Riemann sums by averaging them.

\[ \begin{aligned} \int_a^b f(x)\,dx \approx \sum_{n=0}^{N-1} \frac{1}{2} (f_n + f_{n+1}) \cdot (\Delta x)_n = \sum_{n=0}^{N-1} \frac{1}{2} (f_n + f_{n+1}) \cdot (x_{n+1} - x_n) \\ \end{aligned} \]

If sample points are evenly spaced:

\[ \begin{aligned} \int_a^b f(x)\,dx \approx h \, \bigg(\frac{1}{2}(f_0 + f_N) + \sum_{n=1}^{N-1} f_n \bigg) \\ \end{aligned} \]

where \(\displaystyle h = \frac{b-a}{N} \)

\[ \begin{aligned} i &= 0 + i = \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})\\ &= e^{\frac{i \pi}{2}} \end{aligned} \]

Thanks to Euler! Therefore,

\[ \begin{aligned} i^i &= (e^{\frac{i \pi}{2}})^i = e^{\frac{i^2 \pi}{2}} \\ &= e^{\frac{-\pi}{2}} = \frac{1}{e^{\frac{\pi}{2}}} \\ &\approx 0.2078796 \end{aligned} \]

Furthermore, there is apparently more than one solution, as \(i^i\) can be derived with multiples of \(2 \pi\) in the periodic functions. The solution set can therefore be represented by a sequence, with

\[ \begin{aligned} a_n = e^{-(\frac{\pi}{2} + 2\,n\,\pi)} \end{aligned} \]

where \(n\) is any integer.

For example, for \(n = -1,0,1,\cdots\), etc. \(a_n = 111.3178, 0.2078796, 0.0003882032, \cdots\), etc.

\(\displaystyle (1+x)^a = 1 + ax + a(a-1)\frac{x^2}{2!} + a(a-1)(a-2)\frac{x^3}{3!} + \cdots \)

where \(a\) can be any rational number including fractions and negative numbers !