The comparison test says that if eventually and converges, then converges. If converges, its terms are bounded: . For ,
so comparison with a geometric series proves absolute convergence.
The radius is the number for which the power series converges absolutely for and diverges for . Given , both and converge, so their terms are bounded, say . Then
Thus ; letting gives .
If is differentiable at and at , then is differentiable at and
Write , where , and put . Division by and passage to the limit proves the formula.
The Hermitian conjugate is . A matrix is unitary when , and Hermitian when .
The intermediate value theorem says that a continuous takes every value between and . For , the closed sets where and cannot separate the connected interval; equivalently, taking the supremum of and using continuity gives a point with value .
Set and for . Every interval contains a zero ; continuity on makes take every value between and , yet is discontinuous at .
A monotone function can be discontinuous only by a jump. If it had a jump at , any number strictly between the left and right limits would lie between and but would not be attained, contrary to the hypothesis. Thus it is continuous.
For the last assertion, pass to subsequences with both within of and near , near . For , the intermediate value theorem on the interval joining and gives with ; then . The endpoint cases use the original sequences.

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!
We have two killer features:
  1. 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
  2. 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:
    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 5. . 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.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    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