Associated Topics || Dr. Math Home || Search Dr. Math

### Divergent Infinite Series

```Date: 05/30/2003 at 03:17:12
From: Hong Yew
Subject: Euler's Infinite Series

I bought a book _Praise for The Mathematical Universe_, by William
Dunham, with a chapter on Euler's infinite series. A proof he outlined
that I could not follow is this:

(1)   1 = 1 - a + a - a^2 + a^2 - a^3 + a^3 - ....

1 = 1 and the "a" basically cancelled out.

(2)   1 = (1-a) + (a-a^2) + (a^2 - a^3)...
= (1-a) + a(1-a) + a^2(1-a)....

it factors out (1-a).

(3)   1/1-a = 1 + a + a^2 + a^3....

divides (1-a) on both sides. I'm confused.

1/1-a is clearly between -1 and 0 for any a > 1.

However, the infinite series 1 + a + a^2 + a^3... is infinite for
a > 1.

They should not equate.
```

```
Date: 05/31/2003 at 12:37:39
From: Doctor Rob
Subject: Re: Euler's Infinite Series

Thanks for writing to Ask Dr. Math, Hong Yew.

Operations with divergent infinite series often yield bizarre results.
The series Euler started with diverges. To see why, consider the
sequence of its partial sums. It goes like this:

1, 1-a, 1, 1-a^2, 1, 1-a^3, ...

The odd-numbered partial sums are all 1; the even ones are 1-a^(n/2).
The definition of the sum of any infinite series is the limit of its
sequence of partial sums. If that limit does not exist, then the
series is said to diverge. If the above sequence of partial sums has a
limit, it must be 1 (look at the odd partial sums), but then one would
have to have

lim (1-a^[n/2]) = 1
n->oo

(from the even partial sums), if and only if

0 = lim a^(n/2),
n->oo

if and only if -1 < a < 1.  For |a| >= 1, you have divergence. Then
all series operations such as those performed by Euler are invalid,
and may lead to nonsensical results.  This is what you found.

Feel free to write again if I can help further.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search