Mathematical falacy. ln(1+x) series exapansion:proving 2=1

series expansion: ln(1+x)=1-x^2/2+x^3/3-x^4/4+x^5/5+..........................∞

ln(1+1)=1-1/2+1/3-1/4+1/5..........∞

ln(2)=(1+1/3+1/5+1/7.......)-(1/2+1/4+1/6+1/8........)

ln(2)=(1+1/3+........)-2(1/2+1/4+1/6+1/8...)+(1/2+1/4+1/6+1/8...)

ln(2)=(1+1/2+1/3+1/4+1/5+1/6....)-2(1/2+1/4+1/6...)

ln(2)=(1+1/2+1/3+1/4...)-(1+1/2+1/3+1/4.....)

ln(2)=0

2=e^0

2=1

how?

what is wrong?

pwsnafu
The series ##1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots## is a conditional convergent series, and any conditionally convergent series can be rearranged to obtain any value you want.

Summary: your rearranging is only valid for absolutely convergent series.

I dont know what is a convergent series.what should I refer to.

*conditional convergent series

Step 5 and 6 are wrong since the series is till infinity and you exploited that property. Try it with finite numbers (say, till 1/8) in the same series and you will find out what I mean.

D H
Staff Emeritus
The problem is much earlier than step 5. It's step 3, where he rearranged the series. That is illegal for a conditionally convergent series.

It's particularly illegal here. He rearranged a conditionally convergent series into two divergent series. Informally, he went from a series representing ln(2) (lines 1 and 2) to ∞-∞ (line 3) to ∞-2*∞+∞ (line 4) to ∞+∞-2*∞ (line 5) to ∞-∞ (line 6) to zero (line 7).

Every step from line 3 on is illegal.

Fredrik
Staff Emeritus
Gold Member
I dont know what is a convergent series.what should I refer to.
Any good text with "calculus" in the title, or any text with "analysis" in the title. The idea is pretty simple. A sequence of real numbers is said to be convergent if there's a real number ##x## such that every open interval that contains x also contains all but a finite number of terms of the sequence. That number x is then called the limit of the sequence.

Explanation of the terminology: If the sequence is ##x_1,x_2,\dots##, the ##x_i## with ##i## a positive integer are called the terms of the sequence. An open interval is an interval that doesn't include the endpoints, e.g. the set of all real numbers x such that 0<x<1. The set of all real numbers x such that 0<x≤1 is also an interval, but not an open interval.

For example, the sequence ##1,1/2,1/3,\dots## is convergent because the number 0 has this funny property: For each open interval E that contains 0, there's a positive integer N such that 1/n is in E for all n≥N.

Now consider a sequence ##x_1,x_2,\dots##. For each positive integer ##n##, the number ##s_n=\sum_{k=1}^n x_k## is called the ##n##th partial sum of the sequence. If the sequence of partial sums (i.e. ##s_1,s_2,\dots##) is convergent, we denote its limit by ##\sum_{k=1}^\infty x_k##.

So the "sum" of infinitely many numbers isn't something trivial. It has to be defined. The definition says that we first have to arrange the numbers in a sequence. Then we have to find its limit of partial sums, if it has one at all. Since the "sum" is defined this way, it's not particularly remarkable that it may depend on how you arrange the terms that you want to add up in a sequence.

Fredrik
Staff Emeritus
Gold Member
A question for people who already know calculus/analysis pretty well:

How would you define the term "series"? I mean, ##\sum_{k=1}^\infty x_k## is just a real number, and you wouldn't consider two series ##\sum_{k=1}^\infty x_k## and ##\sum_{k=1}^\infty y_k## to be the same series just because ##\sum_{k=1}^\infty x_k=\sum_{k=1}^\infty y_k##, would you? So it doesn't really make sense to call ##\sum_{k=1}^\infty x_k## a series. Also, it would make phrase "the sum of the series" nonsensical.

What we think of as "the series ##\sum_{k=1}^\infty x_k##" is completely determined by the sequence ##\langle x_k\rangle_{k=1}^\infty##, so it makes sense to call that a series But I don't think I've seen anyone do that.

Maybe "series" should be defined as a function that takes sequences to...what exactly? Only the sequences whose partial sums form a convergent sequence are taken to real numbers.

jbunniii
Homework Helper
Gold Member
A question for people who already know calculus/analysis pretty well:

How would you define the term "series"? I mean, ##\sum_{k=1}^\infty x_k## is just a real number, and you wouldn't consider two series ##\sum_{k=1}^\infty x_k## and ##\sum_{k=1}^\infty y_k## to be the same series just because ##\sum_{k=1}^\infty x_k=\sum_{k=1}^\infty y_k##, would you? So it doesn't really make sense to call ##\sum_{k=1}^\infty x_k## a series. Also, it would make phrase "the sum of the series" nonsensical.
The notation ##\sum_{k=1}^\infty x_k## is shorthand for the sequence of partial sums defined by ##s_n = \sum_{k=1}^n x_k##. Convergence of the series means precisely that the sequence ##(s_n)## converges.

By an abuse of notation, ##\sum_{k=1}^\infty x_k## also means the limit of the sequence ##(s_n)##, when it exists.

Thus ##\sum_{k=1}^\infty x_k## and ##\sum_{k=1}^\infty y_k## are not the same series unless ##x_k = y_k## for all ##k##, but we may still write ##\sum_{k=1}^\infty x_k = \sum_{k=1}^\infty y_k## if they converge to the same limit.

Fredrik
Staff Emeritus
Gold Member
OK, so we use the notation ##\sum_{k=1}^\infty x_k## both for the sequence of partial sums and its limit (if it has one). But what in all this is called a "series"? I guess it must be the sequence of partial sums? Then the phrase "the series ##\sum_{k=1}^\infty x_k##" makes sense, and it still makes sense to talk about the terms of the series, since they can be recovered from from the sequence of partial sums: ##x_{n+1}=s_{n+1}-s_n##.

I just find it a bit odd that this makes the terms "series" and "sequence" synonymous. The words have the same definition, but are still used differently, or rather, the choice of whether to call a given sequence a "sequence" or a "series" just determines what the term "term" will refer to. This is pretty weird, but I guess the alternatives are no better.

jbunniii
Homework Helper
Gold Member
Yes, you can define the series as being precisely the sequence of partial sums. This definition is used, for example, in Apostol's "Calculus."

Spivak sidesteps this terminology and defines what it means for a sequence ##(x_k)## to be summable (the sequence of partial sums converge), and defines the notation ##\sum_{k=1}^\infty x_k## as the limit of the sequence of partial sums. However, he does spend a paragraph acknowledging that the symbol ##\sum_{k=1}^\infty x_k## is often overloaded to mean either the series or the limit, depending on context.

Rudin defines ##\sum_{k=1}^\infty x_k## to be a symbol meaning the sequence of partial sums, and he defines the infinite series to be that symbol.

"Series" and "sequence" are not quite synonymous. You can define sequences with values in any set ##S##: indeed, a sequence is simply a function from ##\mathbb{N}## to ##S##. You can define convergent sequences with values in any metric space, for example. But for a series, you need to be able to add the values, so you need, for example, a normed vector space.

Also, even with real or complex values, there are important notions associated with series, such as absolute versus conditional convergence, which have no counterpart when working with general sequences. We could say things like "if the limit of the sequence of partial sums of the sequence ##|x_n|## exists, then so does the limit of the sequence of partial sums of the sequence ##x_n##", but that is so clumsy to write that it actually obscures the meaning: if ##\sum_{n=1}^\infty |x_n|## converges, then so does ##\sum_{n=1}^\infty x_n##. Since series are ubiquitous throughout analysis, it's best to give them the most efficient notation we can come up with.

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1 person
Homework Helper
What is an infinite series?

The defining of a series as a particular type of sequence is not a new convention. This is from Theory and Application of Infinite Series by Dr. Konrad Knopp (published 1928, page 98). I've included his lead up to the actual definition for reference. Note a few things.
• The language used, while more flowery than that found in modern works, is still precise
• This comes after 97 pages of preliminary work in properties of sequences
• The actual definition states that the series is the symbol used to represent a particular sequence, a sequence of partial sums. That is a different, subtle, emphasis, from what we see today

Infinite series. These are sequences given in the following way. A sequence is at first assigned in any manner (usually by direct indication of the terms), but without being intended itself to form the object of discussion. From it a new sequence is to be deduced, whose terms we now denote by $s_n$, writing

$$s_0 = a_0; s_1 = a_0 + a_1; s_2 = s_0 + s_1 + s_2;$$

and generally

$$s_n = a_0 + a_1 + s_2 + \cdots + a_n \quad (n = 0, 1, 2, 3, \dots)$$

It is the sequence of these numbers which then forms the object
of investigation. For this sequence $\{s_n\}$ we use the symbolical expression

$$a_0 + a_1 + a_2 + \cdots + a_n + \cdots \tag{a}$$

or more shortly

$$a_0 + a_1 + a_2 + \cdots \tag{b}$$

or still more shortly and more expressively;
$$\sum_{n=0}^\infty a_n$$

and this new symbol we call an infinite series; the numbers $s_n$ are called the partial sums or sections of the series. We may therefore state the

Definition An infinite series is a new symbol for a definite sequence of numbers
deducible from it, namely the sequence of partial sums.