To satisfy your curiosity and rescue you from mathematical terms, the simple explanation is: x = 1 + 2 + 4 + 8 + …but for convergent series, like 1/2 + 1/4 + 1/8 + 1/16…

To satisfy your curiosity, rescue you from mathematical terms, the simple explanation is:

x = 1 + 2 + 4 + 8 + …

x = 1+ (2 + 4 + 8 + …)

x = 1+ 2(1 + 2+ 4 + 8…)

x = 1+ 2x

x = -1

This is almost the same as calculating the converging infinite series. But for convergent series, like 1/2 + 1/4 + 1/8 + 1/16…it is easy to visualize and understand, while divergent series is not.

divergence and convergence

convergence series is the series that tends toward a certain number. For example, the convergence series 1/2 + 1/4 + 1/8 + 1/16 +…is obviously approaching a certain limit, i.e. 1, as shown in the geometric diagram below.

We can also use their "partial sum" to distinguish divergence and convergence. As the name implies, "partial sum" is the sum of part of the terms in a sequence. We can use the formula of geometric series to represent the sum of the first n terms of 1/2 + 1/4 + 1/8.

Through this formula, we see that the sum of a converging series seems to be approaching 1, which is more obvious in the function graph.

However, in the case of divergence, the partial sum does not approach a value, but diverges to infinity.

"rules" for calculating divergent series

definition (regular): If the summing method of the series gives the correct answer to the converging series (i.e. the limit of part and sequence), the summing method is regular,

linear

to be linear, and the sum must be allocable and decomposed:

Under linear conditions, terms with sums of equal lengths can be grouped,

Stability

Definition: When the terms can be "extracted" from the sum, the sum method has stability,

Not all -level summing methods meet these conditions (especially stability). Note that most methods of summing series do not work for each series; the goal is to find and use as many interesting and important series addition methods as possible.

linearity combined with stability

x = 1 + 2 + 4 + 8 + …

(1) x = 1+ (2 + 4 + 8 + …)

(2) x = 1+ 2(1 + 2+ 4 + 8…)

x = 1+ 2x

x = -1

Obviously, the limit of this divergence series is infinite, and I got a finite answer. But we can prove that it does conform to linearity and stability,

(1) x = 1+ (2 + 4 + 8 + …)

In (1), we can extract one from the sum, which is equivalent to

So we can say that the sum method is stable.

(2) x = 1+ 2(1 + 2+ 4 + 8…)

For (2) we can propose 2 from the sum formula, which is equal to

This indicates that the series is linear.

With this, I get the answer of 1 + 2 + 4 + 8 + … = -1.

answers all seem strange, but please note that they all represent a continuation of a known geometry series formula,

In calculus, we know that the series converge only when r∈(-1, 1). One way to get the answer to the above example is to use this formula, but substitute it into the value outside the convergence interval, that is, r= 2. Of course, this is not a general summing method, but it does give us an intuitive feeling to let us know where the answer comes from.

Cai ChaluoQiu and

Cai ChaluoQiu's method of summing is as follows: take the limit of the infinite series part and the average value. Assuming a series,

s_k is its kth part sum, then the limit when

k approaches infinity (if it exists) is the limit of the series.

For example, 1-1+1-1+1-1+…, the sum of parts is 1,0,1,0,1…, the series does not converge because the limit of this sequence does not exist. But the average value series of partial sum is,

It converges to 1/2, so this series ( -1+1-1+1-1+…) is the sum of Cai Chaluo? The answer is 1/2.

Caicharo summability allows certain series with oscillating parts and sequences to be "smoothed", but if the partial sum of series becomes infinite (such as the harmonic series), the average value of the partial sum will also reach infinite. The example in this article "1 + 2 + 4 + 8 + …" cannot be asked for peace by Cai Chaluo.

Abel sum

Abel sum involves the limit of power series : If the limit exists, define

This means that if the series converges, then the limit on the right side of the equation above exists and is equal to this sum. Note that 1-1+1-1+1+... is summed by Abel, because

In fact, any series that can be summed by Chai Chaluo is also summed by Abel, and their sum is the same. Therefore, Abel is more summable.

Riemann zeta function regularization

uses the parsing of complex value functions to extend to define some summing methods. The analytical extension of function f is function g, which is defined on a larger set than f, which is differentiable everywhere on the definition domain .

The most inspiring example is the Riemann zeta function

Series converge only when the real part Re(s) of the complex s is greater than 1, but there is a function equation that extends zeta function into a function that is well defined and differentiable everywhere except s=1. This function equation can be calculated as follows,

, so substituted into the series representation of s=-1 to the zeta function, and obtained,

results prove that this is practically used in the calculation of string theory and quantum mechanics one-dimensional Casimir effect.

function

may converge on a complex half plane, but if it can be parsably extended to a function defined as s= -1, the value of the function at -1 can be associated with the sum of the series. Note that this method is stable, but not linear.

Dilicre series regularization

Another concept sometimes called zeta function regularization is Dicrecre series

If f can be parsed and extended to 0, then the value of f(0) is assigned to the right. This is a different approach from regularization of zeta functions, which is linear but unstable.

The sum of divergent series is usually used in physics, such as +2+3+4+.... The general idea is that if a physical situation is described by a function f, which is defined by a series, and it only converges to some value sets that do not include s, then the analytical extension of f g has some larger value sets (including s), which is closely related to f, so that g(s) can have some meaningful physical explanations, even if f(s) is not defined.

Just the study of infinite diverging series can give us some interesting insights, as Leonhard Euler shows us - a profound discovery of the overall mathematics.