Mysterious Holes || Mathematical Analysis || Repeated Series

In this video I will show you a legendary book on mathematical analysis and then we will do some mathematics from this book. The book is called Mathematical Analysis and it was written by Tom Apostol. After discussing the book we will very a statement that is given in the text without proof. The statement has to do with two repeated series that converge but are not equal. The approach we take is a simple one in that we don't assume any prior knowledge.

0:00 The Mysterious Holes
0:29 Introduction
1:12 The Book
6:49 Repeated Series

Here are a few copies of the book.
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[Anonymous-qw]

I used this Apostol book for my second year Analysis Couse for for my Bsc(Hons) Mathematics degree in the UK in the early 1980s. It is one of the books I still have and survived my ex wife who wanted to take all my old college books to a second hand book shop. My copy is a blue paperback. As paperbacks get damaged much more easily it is in a much less good condition than yours. I found it quite terse as well. I also used Dellillo Advanced Calculus with Applications for that course.

[ColonelPanic]

I greedily guard my Apostol books. His work with The Mechanical Universe television series and the MU text got me interested in Mathematics.

[declanfarber]

Apostol and Feynman were contemporaries, and both taught introductory courses at Caltech around that time (ca 1960 give or take.) This is an interesting insight into what it was like to be at that school at that time. The picturing of it is interesting. To be a fly on the wall. Yeah go ahead and smell the binding. :^)

[johnflorio]

We used the Apostol text for Mathematical Analysis in graduate school. It was intimidating at first but thankfully we had a professor who was an outstanding communicator.

[tmendoza]

staples someone gone mad by analysis thinking he can gain a firm understanding
love it great video!

[sanjitdas]

I remember when I bought the first copy (Calculus, vol1) -- in 1999-- red Wiley soft cover-- It was written dedicated to Jane and Stephen! Much later (two years later) I came to know about this book.

[edwardgraham]

I always love your review of mathematical book reviews --- thank you!

[physicshypernova]

I have this one as well. It was helpful in understanding winding numbers when I was preparing for my qualifying presentation for my PhD. The calculus book by Courant was also useful and easier to read than Apostol.

[hussainfawzer]

Though I’ve never seen double summation ever before. I understood your explanation. I feel very happy. 😊

[tamaramacadam]

Oooh yesterday I found this book at my uni library! World Student Series edition, second printing (1965). Definitely going to work through it over the next couple months.

[dd-ufnw]

I'm currently working on apostol cal.1 but i liked it so much that I would also go for his cal2 and mathematical analysis.

[navierstokes]

I think it would be easier to visualize what it's actually happening if you would put a matrix, then one order of the series is column sum and the other order is row sum.

[OleJoe]

I met Tom Apostol once back in the early 1980s at CSULB when I was in grad school.

One of my math professors knew Apostol and invited him to give a lecture on Analytic Number Theory.

Something I knew nothing about and still don't. I went to the lecture just to be there. Got to meet and talk to him afterwards.

[xAssailantx]

"this math book has been damaged..."

Literally the most perfect condition Tom Apostol textbook I've ever seen. Probably worth like 500 dollars.

[John_Smith__]

Yup Tom Apostol Calculus is very very popular. He does have a two volumes book that covers a lot of Calculus. Well written book no doubt.

[mentatphilosopher]

So I thought I would follow along with my second edition from September 73. There has been a good amount of changes. It looks like a lot of the elementary vector calculus went elsewhere; the total page count is smaller at 492. The “repeated series” is now called “integrated series” and the “important case” must no longer be important as it has disappeared completely.

[paulkarch]

When I took calculus, our professor used his book. I found it a little bit hard because I went to a rural high school and I had never seen anything like that before. He begins the book with integral calculus and uses the method of exhaustion known to the Greeks to essentially compute the Riemann integral of a function.

[griffgruff]

This brought back memories. I bought my copy of this book in 1970.

[Zone_Stomper]

You should try your hand at book repair as you have some in need of fixing. If successful, you can show the before and after along with the affiliate link for a book binding and repair kit.

[nice]

I saw it explains Cesaro Summability (3:18) which is quite unexpected for the introductory level.