# 1 + 10 + 100 + 1000 + = -1/9

• Unit
In summary, the conversation discusses the question of whether or not the infinite sum S = 1 + 10 + 100 + 1000 + 10000 + ... converges in different fields, such as the reals or p-adics. The conclusion is that while it does not converge in the reals, it may converge in other fields such as the 2-adics or 5-adics. However, the proof presented may not necessarily hold true if S does not exist.
Unit
$$S = 1 + 10 + 100 + 1000 + 10000 + ...$$

$$10S = 10 + 100 + 1000 + 10000 + 100000 + ...$$

$$S - 10S = (1 + 10 + 100 + 1000 + 10000 + ...) - (10 + 100 + 1000 + 10000 + ...)$$

$$-9S = 1 + (10 - 10) + (100 - 100) + (1000 - 1000) + (10000 - 10000) ...$$

$$-9S = 1 + 0 + 0 + 0 + 0 + 0 ...$$

$$-9S = 1$$

$$S = -1/9$$

What's wrong (or right) with this?

Thanks,
Unit

Also, I believe that sum converges as an ordinary infinite sum in the 2-adics and the 5-adics.

(And, of course, it does not converge as an ordinary infinite sum in the reals!)

I would have [intuitively] expected it to converge in all the p-adics. Am I wrong?

In any other p-adic field, the terms don't converge to zero!

Hurkyl said:
In any other p-adic field, the terms don't converge to zero!

I'm pretty sure that you can't pair up terms in an infinite sum.

Unit said:
$$S = 1 + 10 + 100 + 1000 + 10000 + ...$$
.
.
.
What's wrong (or right) with this?

Thanks,
Unit

Char. Limit said:
I'm pretty sure that you can't pair up terms in an infinite sum.
The way I remember it, proofs like this actually say something like:
If S exists, then S = 1 + 10 + 100 + ...​
So if S does not exist, then the remaining statements do not necessarily hold true.

Redbelly98 said:
If S exists, then S = 1 + 10 + 100 + ...​
So if S does not exist, then the remaining statements do not necessarily hold true.

Brilliant! I had completely forgotten about variables and their related hypothetical syllogisms. Thanks!

## What is the equation 1 + 10 + 100 + 1000 + ... = -1/9?

The equation 1 + 10 + 100 + 1000 + ... = -1/9 is called a geometric series. It is an infinite sum of numbers that follow a pattern where each term is multiplied by a constant factor. In this case, the constant factor is 10. The sum of all the terms in this series is equal to -1/9.

## Why does this equation equal -1/9?

This equation equals -1/9 because it follows the formula for a geometric series: a + ar + ar^2 + ... = a/(1-r), where a is the first term and r is the common ratio. In this case, a = 1 and r = 10. Plugging these values into the formula gives us 1/(1-10) = -1/9.

## How can an infinite sum equal a finite number?

It may seem counterintuitive that an infinite sum can equal a finite number, but in this case, it is due to the pattern and properties of geometric series. As the terms in the series get larger and larger, they also get closer and closer to 0. This means that as we add more and more terms, the sum gets closer and closer to a finite number, in this case, -1/9.

## What is the practical application of this equation?

This equation has practical applications in mathematics, physics, and engineering. It can be used to calculate the sum of infinite series, which is useful in various fields such as calculus, probability, and signal processing. It also has applications in understanding patterns and relationships in nature, such as the growth rate of populations or the behavior of waves.

## Is this equation true for all values of x?

No, this equation is only true for the specific values of x that make up the geometric series. In this case, the equation is only true for x = 1, 10, 100, 1000, and so on. For any other value of x, the sum will be different. This is why it is important to specify the range of values for which an equation is true.

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