Cauchy sequences and absolutely convergent series

In summary: I'm not sure why.In summary, the statement that if X is a normed space, then (b) ⇒ (a) is not always true.
  • #1
Fredrik
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Homework Statement



I want to prove that if X is a normed space, the following statements are equivalent.

(a) Every Cauchy sequence in X is convergent.
(b) Every absolutely convergent series in X is convergent.

I'm having difficulties with the implication (b) ⇒ (a).

Homework Equations



Only some standard definitions.

The Attempt at a Solution



I have no problems with the implication (a) ⇒ (b). I'm including the proof only because it can perhaps provide some inspiration for the proof of the converse.

(a) ⇒(b): Let ##\sum_{k=1}^\infty x_k## be an arbitrary absolutely convergent series in X. For each ##n\in\mathbb Z^+##, define ##s_n=\sum_{k=1}^n x_k## and ##t_n=\sum_{k=1}^n \|x_k\|##. The desired result follows from
$$\|s_n-s_m\|=\left\|\sum_{k=m+1}^n x_k\right\| \leq\sum_{k=m+1}^n\|x_k\| =|t_n-t_m|.$$ The argument is simple: ##(t_n)## is convergent and therefore Cauchy. This and the inequality above together imply that ##(s_n)## is Cauchy. Now (a) implies that ##(s_n)## is convergent.

Here's the part I'm struggling with:

(b) ⇒ (a): Let ##(s_n)## be an arbitrary Cauchy sequence in X. I'm using the notation ##s_n## rather than ##x_n## because I'm going to define a sequence ##(x_n)## such that ##(s_n)## is its sequence of partial sums. We define ##x_1=s_1## and for each ##n\geq 2##, ##x_n=s_n-s_{n-1}##. We have
$$\sum_{k=1}^n x_k =(s_n-s_{n-1})+(s_{n-1}-s_{n-2})+\cdots+(s_2-s_1)+s_1 =s_n.$$ If we can show that ##\sum_{k=1}^\infty x_k## is absolutely convergent, we're done. Unfortunately that doesn't look possible. I think we have to continue something like this: For each ##k\in\mathbb Z^+##, let ##n_k## be a positive integer such that
$$i,j\geq n_k\ \Rightarrow\ \|s_i-s_j\|<2^{-k}.$$ Let ##k\in\mathbb Z^+## be arbitrary. Define ##m_k=n_k+1##. We have
$$\|x_{m_k}\|=\|s_{m_k}-s_{m_k-1}\|=\|s_{n_k+1}-s_{n_k}\|<2^{-k}.$$ Does this help at all? It tells us that ##\sum_{k=1}^\infty x_{m_k}## is absolutely convergent, and therefore convergent. But how does knowing that ##\sum_{k=1}^\infty x_{m_k}## is convergent help us? It means that the sequence ##\big(\sum_{k=1}^p x_{m_k}\big)_{p=1}^\infty## is convergent, but this isn't a subsequence of ##(s_n)##. ##(s_{m_p})_{p=1}^\infty## is a subsequence of ##(s_n)##, but I don't see why it would be convergent.
 
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  • #2
There's a proof here:

https://www.math.ucdavis.edu/~thomases/201A_F10_hw3_sol.pdf
 
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  • #3
Thanks. I understand it now. I need to write ##\|s_{n_{k+1}}-s_{n_k}\|<2^{-k}## instead of ##\|s_{n_k+1}-s_{n_k}\|<2^{-k}##. I tried that earlier, but somehow I convinced myself that if I do this, the sum fails to telescope.
 

Related to Cauchy sequences and absolutely convergent series

1. What is a Cauchy sequence?

A Cauchy sequence is a sequence of real numbers that satisfies the Cauchy criterion, which states that for any positive real number ε, there exists a positive integer N such that for all n, m ≥ N, the absolute value of the difference between the nth and mth term in the sequence is less than ε.

2. How is a Cauchy sequence different from a convergent sequence?

A convergent sequence is a sequence of real numbers that has a limit, meaning that as n approaches infinity, the terms in the sequence get arbitrarily close to the limit. A Cauchy sequence, on the other hand, does not necessarily have a limit, but rather satisfies a specific criterion of closeness between terms in the sequence.

3. What does it mean for a series to be absolutely convergent?

A series is absolutely convergent if the sum of the absolute values of its terms is finite. This means that even if the terms themselves may not converge, the sum of their absolute values does.

4. How can you determine if a series is absolutely convergent?

A series can be determined to be absolutely convergent by applying the Cauchy condensation test, which states that if the series of the powers of 2 multiplied by the terms in the original series (i.e. 2^n * a_n) converges, then the original series is absolutely convergent.

5. What is the relationship between Cauchy sequences and absolutely convergent series?

Cauchy sequences and absolutely convergent series are related in that a series is absolutely convergent if and only if its sequence of partial sums is a Cauchy sequence. This means that the sum of the absolute values of the terms in an absolutely convergent series will converge as well.

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