1 − 1 + 1 − 1 + … = ???

[⛄]

[windjammer]

Other: no limit.

[Franzl]

Other: no limit.

Correct.

A sum has a limit, then and only then, when the respective series of numbers converges towards 0.

[muon2]

Other: no limit.

Correct.

A sum has a limit, then and only then, when the respective series of numbers converges towards 0.

You may be confusing the series and the sequence. The set of numbers {1, -1, 1, -1, ...} is a sequence. The series is the sum of the sequence as in the OP. That particular series even has a name, Grandi's series, and it is divergent.

The sequence {1, 1/2, 1/4, 1/8, ...} converges to 0, but the series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2. However, there are sequences that converge to zero, but form divergent series such as {1, 1/2, 1/3, 1/4, 1/5, ...}.

[Franzl]

Other: no limit.

Correct.

A sum has a limit, then and only then, when the respective series of numbers converges towards 0.

You may be confusing the series and the sequence. The set of numbers {1, -1, 1, -1, ...} is a sequence. The series is the sum of the sequence as in the OP. That particular series even has a name, Grandi's series, and it is divergent.

The sequence {1, 1/2, 1/4, 1/8, ...} converges to 0, but the series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2. However, there are sequences that converge to zero, but form divergent series such as {1, 1/2, 1/3, 1/4, 1/5, ...}.

I had a translation problem there. I thought series was the translation for "Folge", and sum for "Reihe".

Yes, I understand. Using the correct terminology now, a series cannot be convergent if the respective sequence does not converge to 0. I didn't mean to imply that this meant that all series of sequences that converge to 0 are also convergent, but merely that it is a necessary condition for a convergent series.

(Assuming I understand that correctly...with my limited undergraduate knowledge from math lectures in my economics program. )

[Foucaulf]

Other: it is Cesàro summable to 1/2.

Given you pulled the earlier trick with the analytic continuation, I don't expect anything different here.

[muon2]

Unfortunately series is not a particularly helpful word to describe the mathematical concept. Sometimes people think of the series as the sequence of partial sums as a way to relate the word series to its conventional meaning.

[Hash]

[Filuwaúrdjan]

k;fk;dzok;oikh;ohk;zh9559595alhmhkly'\03q44

[DemPGH]

k;fk;dzok;oikh;ohk;zh9559595alhmhkly'\03q44

LOL! Belongs in the good post gallery. (Is there a LMAO emoticon?)

[windjammer]

"La série de Grandi est divergente "
So yeah, it has officially no limit, so even if the result is 1/2, I wasn't wrong .

[Citizen (The) Doctor]

You didn't happen to take a Calculus test today did you?

[⛄]

Hint: This series has two possible solutions.

[⛄]

You may be confusing the series and the sequence. The set of numbers {1, -1, 1, -1, ...} is a sequence. The series is the sum of the sequence as in the OP. That particular series even has a name, Grandi's series, and it is divergent.

The sequence {1, 1/2, 1/4, 1/8, ...} converges to 0, but the series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2. However, there are sequences that converge to zero, but form divergent series such as {1, 1/2, 1/3, 1/4, 1/5, ...}.

An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders half a beer. The third one orders a fourth of a beer. The bartender stops them, pours two beers and says, "You guys should know your limits."

[PiMp DaDdy FitzGerald]

Hint: This series has two possible solutions.

There is an infinite amount of solutions, if the series is interpreted as indeterminant.

[Smid]

Saw this great YouTube video on this topic the other week. I'm now subscribed to their channel.

[muon2]

Hint: This series has two possible solutions.

Not true. There are two accumulation points 0 and 1, but there is no conventional solution because the series diverges. Alternatively there are a wide variety of well-defined summation techniques; for example Cesaro summation gives a single solution to the series equal to 1/2.

[True Federalist (진정한 연방 주의자)]

The odd thing about Cesaro is that the order of what you sum matters.

Each of the below has the same number of + 1's and - 1's and a partial sum of 0 every fourth entry yet you get five different results

+ 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... = +½

- 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - ... = -½

+ 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + ... = 1

+ 1 - 1 - 1 + 1 + 1 - 1 - 1 + 1 + ... = 0

- 1 - 1 + 1 + 1 - 1 - 1 + 1 + 1 -  ... = -1

I'm not certain, but it looks like you can generate any rational number as the Cesaro sum with an appropriate ordering of the +1's and -1's that periodically has a partial sum of 0.  You could probably even generate irrational numbers with a defined aperiodic ordering of the +1's and -1's that had infinitely many partial sums of 0.  I wonder if anyone has come up with an algorithm to generate such an ordering for any desired real number.  Would be neat if they have.

[H.E. VOLODYMYR ZELENKSYY]

Saw this great YouTube video on this topic the other week. I'm now subscribed to their channel.

Numberphile? Yeah, they're good.

[muon2]

The odd thing about Cesaro is that the order of what you sum matters.

Each of the below has the same number of + 1's and - 1's and a partial sum of 0 every fourth entry yet you get five different results

+ 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... = +½

- 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - ... = -½

+ 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + ... = 1

+ 1 - 1 - 1 + 1 + 1 - 1 - 1 + 1 + ... = 0

- 1 - 1 + 1 + 1 - 1 - 1 + 1 + 1 -  ... = -1

I'm not certain, but it looks like you can generate any rational number as the Cesaro sum with an appropriate ordering of the +1's and -1's that periodically has a partial sum of 0.  You could probably even generate irrational numbers with a defined aperiodic ordering of the +1's and -1's that had infinitely many partial sums of 0.  I wonder if anyone has come up with an algorithm to generate such an ordering for any desired real number.  Would be neat if they have.

You are correct, Cesaro summation relies on the specific ordering of the sequence. Infinite series do not in general obey the associative or commutative properties of finite addition. Your example is a good illustration of that.

[dead0man]

[Starbucks Union Thug HokeyPuck]

It equals 1/2.  

EDIT: And yes, Numberphile is an incredible channel.  One of the best on Youtube.