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tends to one rather than zero. The term test for divergence therefore proves the series diverges. A ratio limit equal to one would be inconclusive and is unnecessary here.

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 2 / i / Solution by

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For

n \geq 1

ln n \leq n

and

2 n^{3} + 1 \geq 2 n^{3}

. Hence

0 \leq ln n / (2 n^{3} + 1) \leq 1/ (2 n^{2})

. The comparison p-series has exponent two and converges, so the series converges by the comparison test.

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 11 / c / Solution by

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On an interval with

sin x \neq = 0

, divide by

sin x

and use integrating factor

1/ sin x

. Then

(\frac{y}{s i n x})^{'} = \frac{c o s ^{3} x}{s i n ^{2} x} .

(1)

Put

u = sin x

in the integral:

\int cos^{3} x d x / sin^{2} x = \int (u^{- 2} - 1) d u = - u^{- 1} - u + C

. Multiplying back gives

y (x) = - 1 - sin^{2} x + C sin x .

(2)

Although the derivation divided by

sin x

, this final expression is smooth and directly satisfies the original undivided equation at every real

x

, including multiples of

π

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The initial value is positive, so set

z = y^{- 2}

. The chain rule gives

z^{'} = - 2 y^{- 3} y^{'} = - 2 (y^{- 2} - 1)

, making

z^{'} + 2 z = 2

. Its solution is

z = 1 + C e^{- 2 x}

. Since

y (0) = 1/2

z (0) = 4

and

C = 3

. Choosing the positive branch gives

y (x) = \frac{1}{1 + 3 e ^{- 2 x}}, x \in R .

(1)

It satisfies the initial condition and approaches the stable positive equilibrium one as

x \to \infty

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 11 / a / Solution by

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First retain the equilibrium solution

y = 0

. For

y \neq = 0

and

x \neq = \pm 1

, separation of variables gives

\int y^{- 2} d y = \int x / (1 - x^{2}) d x

, hence

- 1/ y = - \frac{1}{2} ln ∣1 - x^{2} ∣ + C

. Renaming the constant yields

y = 0, or y = \frac{2}{ln ∣1 - x ^{2} ∣ + C} .

(1)

A nonzero solution is considered on a regular interval avoiding

x = \pm 1

and zeros of its denominator. The zero solution itself extends through both singular points of the divided equation.

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 10 / c / ii by

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 10 / b / iii by

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The modulus equation says that

z

has fixed distance eight from

- 1 - i

. Thus its locus is a circle of radius eight centered at $(- 1, - 1)$ , with Cartesian equation

(x + 1)^{2} + (y + 1)^{2} = 64

Figure 1.
The original circle, a horizontal stretch, a translation and a quarter-turn
. Original complex-plane loci. The map $u = 2 x + i y$ stretches only the horizontal coordinate; $v = z + 4 + 3 i$ translates the circle; $w = i v$ rotates that translated circle about the origin. Marked dots show each center. **Details**.

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Take the real part of De Moivre's theorem. Put

c = cos θ

and use

sin^{2} θ = 1 - c^{2}

cos 5 θ = c^{5} - 10 c^{3} (1 - c^{2}) + 5 c (1 - c^{2})^{2} = 16 cos^{5} θ - 20 cos^{3} θ + 5 cos θ .

(1)

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 1 / 9 / c / Solution by

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Two applications of integration by parts give

I = cosh x sin x - \int sinh x sin x d x

and

\int sinh x sin x d x = - sinh x cos x + I

. Thus an antiderivative is

[cosh x sin x + sinh x cos x] /2

. Evaluating its endpoints gives

\int_{0}^{π /2} cosh x cos x d x = \frac{1}{2} cosh \frac{π}{2} .

(1)

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Adding and subtracting the exponential function power series cancels the appropriate odd or even terms:

sinh x = n = 0 \sum \infty \frac{x ^{2 n + 1}}{( 2 n + 1 )!} = x + \frac{x ^{3}}{3 !} + \dots, cosh x = n = 0 \sum \infty \frac{x ^{2 n}}{( 2 n )!} = 1 + \frac{x ^{2}}{2 !} + \dots .

(1)

Both series converge absolutely for every complex number

x

, as follows from the ratio test.

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The exponential function definitions of the hyperbolic functions and Euler's formula give

sinh (i x) = \frac{e ^{i x} - e ^{- i x}}{2} = i sin x, cosh (i x) = \frac{e ^{i x} + e ^{- i x}}{2} = cos x .

(1)

Dividing these identities wherever

cos x \neq = 0

gives

tanh (i x) = i tan x

. The two undivided identities hold for every complex number

x

; the quotient identity excludes its poles.

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Write

1/ f (x) = b_{0} + b_{1} x + b_{2} x^{2} + O (x^{3})

and multiply by

a_{0} + a_{1} x + a_{2} x^{2} + O (x^{3})

. The product equals one, so its coefficients give

a_{0} b_{0} = 1

a_{0} b_{1} + a_{1} b_{0} = 0

and

a_{0} b_{2} + a_{1} b_{1} + a_{2} b_{0} = 0

. Since

a_{0} = f (0) \neq = 0

\frac{1}{f ( x )} = \frac{1}{a _{0}} - \frac{a _{1}}{a _{0}^{2}} x + \frac{a _{1}^{2} - a _{0} a _{2}}{a _{0}^{3}} x^{2} + O (x^{3}) .

(1)

This is the initial part of the general reciprocal of a power series recurrence.

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

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