Proof of Cauchy Sequence for $\{a_n\}$ Defined by $f(x)$

In summary, the sequence $\{a_n\}$ is a Cauchy sequence. $\lim a_n$ is equal to the sum of the infinite series: $e^{-k-1}$.
  • #1
alexmahone
304
0
Suppose $f(x)$ is continuous and decreasing on $[0, \infty]$, and $f(n)\to 0$. Define $\{a_n\}$ by

$a_n=f(0)+f(1)+\ldots+f(n-1)-\int_0^n f(x)dx$

(a) Prove $\{a_n\}$ is a Cauchy sequence directly from the definition.

(b) Evaluate $\lim a_n$ if $f(x)=e^{-x}$.
 
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  • #2
We can write $a_n=\sum_{k=0}^{n-1}\int_k^{k+1}(f(k)-f(x))dx$. Does it help you?
 
  • #3
girdav said:
We can write $a_n=\sum_{k=0}^{n-1}\int_k^{k+1}(f(k)-f(x))dx$. Does it help you?

Do you mean to interpret it geometrically as $T_1+T_2+\ldots+T_n$ as in the following figure?

https://www.physicsforums.com/attachments/36
 
  • #4
Yes, it will give you the idea, and we can see what $a_n$ represents. Now you have to show analytically that $\{a_n\}$ is a Cauchy sequence.
 
  • #5
girdav said:
Now you have to show analytically that $\{a_n\}$ is a Cauchy sequence.

I think I can show it geometrically.

$a_m-a_n=T_{n+1}+T_{n+2}+\ldots+T_m$ for $m>n$

Given $\epsilon>0$,

$f(n+1)<\epsilon$ for $n\gg 1$

By moving all the "triangles" from $T_{n+1}$ to $T_m$ horizontally to the left into the rectangle of base 1 and height $f(n+1)$ (as shown in the figure), we see that

$T_{n+1}+T_{n+2}+\ldots+T_m<f(n+1)<\epsilon$

Does that look ok?
 
  • #6
(b) $a_n=\sum_{k=0}^{n-1}\int_k^{k+1}(f(k)-f(x))dx$

$=\sum_{k=0}^{n-1}\int_k^{k+1}(e^{-k}-e^{-x})dx$

$=\sum_{k=0}^{n-1}\int_k^{k+1}(xe^{-k}+e^{-x})$

$=\sum_{k=0}^{n-1}\int_k^{k+1}((k+1)e^{-k}+e^{-k-1}-ke^{-k}-e^{-k})$

$=\sum_{k=0}^{n-1}e^{-k-1}$

$\lim a_n=\sum_{k=0}^\infty e^{-k-1}$

$=\frac{e^{-1}}{1-e^{-1}}$

$=\frac{1}{e-1}$
 
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  • #7
Alexmahone said:
Suppose $f(x)$ is continuous and decreasing on $[0, \infty]$, and $f(n)\to 0$. Define $\{a_n\}$ by

$a_n=f(0)+f(1)+\ldots+f(n-1)-\int_0^n f(x)dx$

(a) Prove $\{a_n\}$ is a Cauchy sequence directly from the definition.

(b) Evaluate $\lim a_n$ if $f(x)=e^{-x}$.

The first thing we need is that for a non-negative decreasing function:

\[ f(k+1) \le \int_k^{k+1} f(x)\;dx \le f(k) \]

Hence for \(m>n\):

\[ \sum_{k=n+1}^{m}f(k) \le \int_n^m f(x)\;dx\le \sum_{k=n}^{m-1}f(k) \]

Now :

\[ a_m-a_n=\sum_{k=n}^{m-1}f(k) -\int_n^mf(x)\;dx \]

So:

\[ \sum_{k=n}^{m-1}f(k) -\sum_{k=n}^{m-1}f(k) \le a_m-a_n \le \sum_{k=n}^{m-1}f(k) -\sum_{k=n+1}^{m}f(k) \]

simplifying:

\[ 0 \le a_m-a_n \le f(n)-f(m) \]

Hence \( \displaystyle \lim_{n,m\to \infty}|a_m-a_n|=0 \)

You will need to check that the above is correct, as it is too easy for the odd index to go wrong here and there, as it has done numerous times while constructing this post (Angry)

CB
 
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FAQ: Proof of Cauchy Sequence for $\{a_n\}$ Defined by $f(x)$

What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers that converges to a limit, meaning that the terms in the sequence get closer and closer to a certain value as the sequence progresses.

How is a Cauchy sequence defined?

A Cauchy sequence is defined as a sequence of numbers in which for every positive real number, there exists an index after which all subsequent terms in the sequence are within that distance of each other.

How can a Cauchy sequence be proven?

A Cauchy sequence can be proven using the Cauchy criterion, which states that a sequence is Cauchy if and only if for every positive real number, there exists an index after which all subsequent terms in the sequence are within that distance of each other.

What is the importance of proving a sequence is Cauchy?

Proving a sequence is Cauchy is important because it shows that the sequence converges to a limit, meaning that the terms in the sequence get closer and closer to a certain value as the sequence progresses. This is a fundamental concept in calculus and analysis.

What role does the function $f(x)$ play in the proof of a Cauchy sequence?

The function $f(x)$ defines the sequence $\{a_n\}$ and is used to evaluate the terms in the sequence. It is important in the proof because it helps to show that the terms in the sequence are getting closer and closer to a limit as the sequence progresses.

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