Conditional and absolute convergence (of series)

[Damidami]

I was reading this article of wikipedia:

Conditional and absolute convergence

It says:

"An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long."

Is that a characterization of absolute convergent sequences? (Does all conditional convergent have infinite long path of partial sums)?

I'm not sure I understand this. I would like to see an example of a conditionally convergent series (of real numbers, not complex numbers if possible), and how is the length of this path related to the non-absolute convergence of the series.

In other words: Is the definition of conditional convergence equivalent to "the length of the path of partial sums diverge"? If true, any proof?

Thanks!

 

[mathman]

Let a1 + a2 + a3 + ... be a series. It is absolutely convergent if |a1| + |a2| + |a3| + ... converges. If it is not absolutely convergent, but it converges, then it is called conditionally convergent.

Example: 1 - 1/2 + 1/3 - 1/4 + ... = ln2. However, the series not absolutely convergent.

 

[Damidami]

Let a1 + a2 + a3 + ... be a series. It is absolutely convergent if |a1| + |a2| + |a3| + ... converges. If it is not absolutely convergent, but it converges, then it is called conditionally convergent.

Example: 1 - 1/2 + 1/3 - 1/4 + ... = ln2. However, the series not absolutely convergent.

Hi mathman,
Thanks, but I already knew that.
In fact, I run those comands in maple:

> evalf(ln(2));
print(`output redirected...`); # input placeholder
0.6931471806


> evalf(seq(sum(a(n), n = 1 .. k), k = 1 .. 10));
print(`output redirected...`); # input placeholder
1., 0.5000000000, 0.8333333333, 0.5833333333, 0.7833333333,

0.6166666667, 0.7595238095, 0.6345238095, 0.7456349206,

0.6456349206

(It can be "seen" that it is going to ln(2))

> evalf(seq(sum(abs(a(n+1)-a(n)), n = 1 .. k), k = 1 .. 10));
print(`output redirected...`); # input placeholder
1.500000000, 2.333333333, 2.916666667, 3.366666667, 3.733333333,

4.042857143, 4.310714286, 4.546825397, 4.757936508, 4.948845599

(It seems that this sequence diverges to infinity)

What I don't know is a proof that the length of the path been infinite implies that the series converges only conditionally.
Does it has something to do with bounded variation functions?

 

[homeomorphic]

What I don't know is a proof that the length of the path been infinite implies that the series converges only conditionally.

That is part of the DEFINITION of conditional convergence. It is only a geometric interpretation of the definition given above, in terms of taking the absolute values and adding them up. No proof necessary. It is a just a visualization of the definition, no more.

When you take the absolute values, you compute the length of each term in the sum.

Say you have a series a0 +a1+a2+...

Picture each term in the series as a vector. If you add (put them tail to tip) all the vectors together in a finite sum and take the length of the result, it will be shorter than adding all the lengths of the vectors together. This is part of why absolute convergence is stronger than conditional convergence.

By the way, you have to prove that series diverge. Computing partial sums is not good enough. Some series diverge very slowly and some series seem to diverge rapidly, but eventually they slow down and converge.

 

[Damidami]

Hi homeomorphic,
Thanks! That makes it a lot clearer.
My mistake was I thougt the sequence a_1,a_2, \ldots a_n already as the "path" and thought the length of vectors they were talking about were "from one point to the next", that is v_i = a_{i+1} - a_i
That explains the code I run in maple. And yes, I know how to use the integral test to prove the divergence of the armonic series