Infinite Series Manipulations

I would suggest you read the book "An introduction to the Theory of Infinite Series" by Thomas Bromwich.

P.S.: There is a tale that goes something like Ramanujan was advised by British Mathematicians to read this book when he wanted to prove that the sum of a series of positive terms is negative ( I neither remember the series, nor Ramanujan's "sum" but all I could remember is it was divergent").

An online version, downloadable in many different formats, is here on the archive.com .

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Read Comment below by Sivaram where he points out both the series and the sum!

Infinite series bugged mathematicians and thinkers for a long time. You can read a little about Zeno's Paradoxes, particularily, the dichotomy paradox.

I guess important questions were

1. Why would an infinite amount of positive quantities be finite?
2. How can we assingn a sum to an infinite amount of numbers, and to a divergent series?
3. Do the known arithmetics work for infinite series?
4. If the sum is finite, how can we find the exact value of the sum?
5. Does the order of the sum matter?

Many of this questions were answered, and some criteria was established such as

1. What an infinite series is.
2. Converge and divergence of a series. D'Alabert, Cauchy, Raabe, Kummer and Gauss provided several methods to sort this out.
3. The concept of absolute convergence and conditional convergence
4. The uniform convergence of a series of functions.
5. The sum of alternating series.

You will find particularily helpful to read about conditional and aboslute convergence: basically, if a series is absolutely convergent, we can manipulate it with the ordinary arithmetic, but if it is conditionally convergent, we can make it sum to any number we want, or make it diverge to infinity. For a divergent series, the normal arithmetic fails, basically, because we're thinking about infinity as a number, when it isn't one. For example, let $S$ denote the infinite sum:

$$S = 1+2+4+8+\cdots$$

We can note that

$$S = 1+2(1+2+4+\cdots)$$

that is

$$S = 1+2S$$

which means

$$-S = 1 $$ or

$$S=-1 \text{ (!)}$$

In the usual sense of arithmetic and algebra, if we are summing infinite many positive numbers, we'd at least expect the sum to be positive, but if we apply the usual rules of arithmetic, we find it's actually $-1$. What does this tell us?

1. $\infty$ is not a number.
2. If we want to make sums like $S$ meaningful, we need to develop a new theory for infinite divergent sums.