By N. A. Sanin

ISBN-10: 0821813684

ISBN-13: 9780821813683

ISBN-10: 0821815717

ISBN-13: 9780821815717

ISBN-10: 0821832662

ISBN-13: 9780821832660

ISBN-10: 1114737356

ISBN-13: 9781114737358

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**Additional resources for Constructive Real Numbers and Function Spaces **

**Example text**

Also, what about the series with the reciprocals of the squares of the integers: J_ J_ J_ J_ J_ l 2 + 2 2 + 3 2 + 4 2 + 5 2 + ? ) Infinity and Infinite Series 9 The Two Halves of the Harmonic Series 1 1 1 1 1 1 3 5 7 9 —^ oo 1 1 1 1 1 - +- +- +- +— + 2 4 6 8 10 —^ oo See Proof 14 —» (page 165) IF THE SUM of the reciprocals of all integers tends to infinity, what about the sum of the series which consists of only some of the terms? What about the sum of the reciprocals of all the odd integers? Or the even integers?

It's just that I have not seen it before in the mathematics literature that I've read so far. This incident shows that there are still "eureka" moments when one discovers something by one's own effort. It does not matter, as in the case of Liebniz, that someone else had discovered it before. I told my granddaughter Rebecca after the discovery. Now she calls the series "Grampa's Series", and writes it out faithfully as another of her "passwords". 7i-series 29 The Euler Series JL JL J_ J_ _L J_ l2~+22~+32 + 4 2 + 5 2 + 62 6 See Proof 41 —> (page 214) IN THE 17TH and 18 th century, mathematicians were seeking ways of summing infinite series of all sorts and patterns.

1 2 3 4 5 6 , 1 1 1 1 1 4. - + - + - + - + - + ••• -^°o 1 3 5 7 9 1 1 1 1 1 r 5. — + — + - + - + — + ••• -^°° 2 4 6 8 10 . 1 1 1 1 1 6. - + - + - + - + — +••• =2 1 2 4 8 16 1 1 1 1 1 8. 1 + - + — + — + — +••• 1! 2! 3! 4! ) 9. i - I + I - i + I - . . ) 1 2 3 4 5 10. I - I + I - I + I - . . ) 4 n-series Know you of this fair work? Beyond the infinite and boundless ... William Shakespeare (1564-1616) Mathematics possesses not only truth, but some supreme beauty Bertrand Russell (1872-1970) 7i-series 25 The Liebniz-Gregory Series 1 1 1 1 1 1 1 + + +— 1 3 5 7 9 11 13 _ n ~7 See Proof 40 -> (page 213) LET US NOW make a more detailed acquaintance with the LiebnizGregory series, which we saw in our introduction to "convergent series".

### Constructive Real Numbers and Function Spaces by N. A. Sanin

by Joseph

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