Express the limit as a definite integral

In summary: R(f,P,c)=\sum_{i=1}^n f(c_i)\Delta x_i$$In summary, the Riemann sum of ##f## on ##[a,b]## corresponding to partition ##P## and tags ##c## is the limit of a sequence of Riemann sums as the number of subintervals of ##P## increases to infinity.
  • #1

mcastillo356

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TL;DR Summary
Assumed continuity implies integrability, can´t see the Riemann sum in this exercise
Hi, PF, there goes the definition of General Riemann Sum, and later the exercise. Finally one doubt and my attempt:

(i) General Riemann Sums
Let ##P=\{x_0,x_1,x_2,\cdots,x_n\}##, where ##a=x_0<x_1<x_2<\cdots<x_n=b##, be a partition of ##[a,b]##, having norm ##||P||=\mbox{max}_{1<i<n}\Delta x_i##. In each subinterval ##[x_{i-1},x_i]## of ##P## pick a point ##c_i## (called a tag). Let ##c=(c_1,c_2,\cdots,c_n)## denote the set of these tags. The sum

$$R(f,P,c)=\sum_{i=1}^n f(c_i)\Delta x_i$$
$$\qquad{=f(c_1)\Delta x_1+f(c_2)\Delta x_2+f(c_3)\Delta x_3+\cdots +f(c_n)\Delta x_n+}$$

is called the Riemann sum of ##f## on ##[a,b]## corresponding to partition ##P## and tags ##c##.
(...) For any choice of the tags ##c##, the Riemann sum ##R(f,P,c)## satisfies

$$L(f,P)\leq R(f,P,c)\leq U(f,P)$$

Therefore, if ##f## is integrable on ##[a,b]##, then its integral is the limit of such Riemann sums, where the limit is taken as the number ##n(P)## of subintervals of ##P## increases to infinity in such a way that the lengths of all the subintervals approach zero. That is,

$$\displaystyle\underset{||P||\to 0}{\lim_{n(P)\to\infty}}{R(f,P,c)}=\int_{a}^{b} f(x) \, dx$$

EXAMPLE 4 Express the limit $$\displaystyle\lim_{n\to\infty}{\sum_{i=1^{n}}{ \displaystyle\frac{2}{n}}\left(1+\displaystyle\frac{2i-1}{n}\right)^{1/3} }$$
as a definite integral.

Solution We want to interpret the sum as a Riemann sum for ##f(x)=(1+x)^{1/3}##. The factor ##2/n## suggests that the interval of integration has length ##2## and is partioned into ##n## equal subintervals, each of length ##2/n##. Let ##c_i=(2i-1)/n## for ##i=1,2,3,\cdots,n##. As ##n\to\infty##, ##c_1=1/n\to 0## and ##c_n=(2i-1)/n \to 2##. Thus, the interval is ##[0,2]## and the points of the partition are ##x_i=\frac(2i)/{n}##. Observe that ##x_{i-1}=(2i-2)/n<c_i<2i/ n=x_i##for each ##i##, so that the sum is indeed a Riemann sum for ##f(x)## over ##[0,2]##. Since ##f## is continous on that interval, it is integrable there, and

$$\displaystyle\lim_{n\to\infty}{\sum_{i=1^{n}}{ \displaystyle\frac{2}{n}}\left(1+\displaystyle\frac{2i-1}{n}\right)^{1/3} }=\int_{0}^{2}{(1+x)^{1/3}} \, dx$$

(iii) Question: how can I state the following equivalence?

$$x_{i-1}=(2i-2)/n<c_i<2i/ n=x_i$$
$$\Longleftrightarrow$$
$$L(f,P)\leq R(f,P,c)\leq U(f,P)$$

Attempt to solve: I can`t. ##c_i## is a tag set; any partition ##P##; but I can say: if ##f## is continous, I can assert the correspondence.

Regards!
 
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  • #2
You didn't define the lower and upper Darboux sums ##L(f,P), U(f,P)## but let's use the standard definitions from here. The relationship follows automatically from the definition since ##t_i\in[x_i,x_{i+1}]## gives us:
$$\inf_{t\in [x_i,x_{i+1}]} f(t) \le f(t_i) \le \sup_{t\in [x_i,x_{i+1}]} f(t)$$
We then sum the inequality over ##i## to get the result.

BTW the definition of the Riemann integral in the OP is no good, as it requires the axiom of choice (because it requires choosing a set of tags in every one of the uncountably infinitely many possible partitions), and Riemann integrability does not need that axiom. See the second definition in this section for an intuitive, workable and robust definition of a Riemann integral.
 
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  • #3
andrewkirk said:
You didn't define the lower and upper Darboux sums ##L(f,P), U(f,P)## but let's use the standard definitions from here. The relationship follows automatically from the definition since ##t_i\in[x_i,x_{i+1}]## gives us:
$$\inf_{t\in [x_i,x_{i+1}]} f(t) \le f(t_i) \le \sup_{t\in [x_i,x_{i+1}]} f(t)$$
We then sum the inequality over ##i## to get the result.
$$L(f,P)=\sum_{i=0}^{n-1}\inf_{t\in [x_{i},x_{i+1}]}=\sum_{i=0}^{n-1}\sup_{t\in [x_{i},x_{i+1}]}=U(f,P)$$

This is my first step, but shouldn't be the last. Working on it.
andrewkirk said:
BTW the definition of the Riemann integral in the OP is no good, as it requires the axiom of choice (because it requires choosing a set of tags in every one of the uncountably infinitely many possible partitions), and Riemann integrability does not need that axiom.
Brilliant remark. Unfortunately, the link to Wikipedia is only a hint to me.
 
  • #4
mcastillo356 said:
TL;DR Summary: Assumed continuity implies integrability, can´t see the Riemann sum in this exercise

EXAMPLE 4 Express the limit
$$ \displaystyle \lim_{n\to\infty}{\sum_{i=1^{n}}{ \displaystyle\frac{2}{n}}\left(1+\displaystyle\frac{2i-1}{n}\right)^{1/3} }$$

as a definite integral.

You have a typo in your LaTeX for the summation: ##\displaystyle \sum_{i=1^{n}}## .

In particular, I wondered what this meant: ##\displaystyle i=1^{n}##

It's simply the result of a brace, ##\rbrace## , in the wrong place. \sum_{i=1^{n}} should be \sum_{i=1}^{n} .

Giving ##\displaystyle \quad \quad \sum_{i=1}^{n}##
 
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  • #5
SammyS said:
It's simply the result of a brace, ##\rbrace## , in the wrong place. \sum_{i=1^{n}} should be \sum_{i=1}^{n} .
In fact, braces aren't even needed at all around that upper limit, n. Braces are needed only when the thing (exponent, subscript, fraction numerator or denominator, radicand, etc.) is more than 1 character.
This will work exactly the same \sum_{i=1}^n .
 
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  • #6
Hi, PF
I will quote the textbook to see if I have solved the question, that is, how can I state first ##x_{i-1}=\frac{(2i-1)}{n}<c_i<\frac{2i}{n}=x_i## for each ##i##, and observe that the sum is indeed a Riemann sum for ##f(x)## over ##[0,2]##:
"Note in Figure 5.13 that ##R(f,P,c)## is a sum of signed areas of rectangles between the ##x##-axis and the curve ##y=f(x)##. For any choice of the tags ##c##, the Riemann sum ##R(f,P,c)## satisfies

$$L(f,P)\leq R(f,P,c)\leq U(f,P)$$

Therefore, if ##f##is integrable on ##[a,b]##, then its integral is the limit of such Riemann sums, where the limit is taken as the number ##n(P)## of subintervals of ##P## increases to infinity in such a way that the lengths of all the subintervals approach zero. That is,

$$\underset{||P||\to 0}{\lim_{n(P)\to\infty}}R(f,P,c)=\int_a^b\,f(x)dx$$

With my words: this paragraph states that tagged partitions implie, likewise partitions, bounded Riemann sums; thus, integrability. Right?

Express.jpg Figure 5.13
 
  • #7
mcastillo356 said:
Hi, PF
I will quote the textbook to see if I have solved the question, that is, how can I state first ##x_{i-1}=\frac{(2i-1)}{n}<c_i<\frac{2i}{n}=x_i## for each ##i##, and observe that the sum is indeed a Riemann sum for ##f(x)## over ##[0,2]##:
"Note in Figure 5.13 that ##R(f,P,c)## is a sum of signed areas of rectangles between the ##x##-axis and the curve ##y=f(x)##. For any choice of the tags ##c##, the Riemann sum ##R(f,P,c)## satisfies

$$L(f,P)\leq R(f,P,c)\leq U(f,P)$$

Therefore, if ##f##is integrable on ##[a,b]##, then its integral is the limit of such Riemann sums, where the limit is taken as the number ##n(P)## of subintervals of ##P## increases to infinity in such a way that the lengths of all the subintervals approach zero. That is,

$$\underset{||P||\to 0}{\lim_{n(P)\to\infty}}R(f,P,c)=\int_a^b\,f(x)dx$$
With my words: this paragraph states that tagged partitions implie, likewise partitions, bounded Riemann sums; thus, integrability. Right?
What you've written doesn't make sense to me.
The "tags" of a partition are merely the set of points ##c_i## in the various subintervals.

When you say "thus integrability," that's trivially true since the assumption is that "if f is integrable on [a, b]..." This is like saying "if x = 2 <whole bunch of other equations> then x = 2."
 
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  • #8
Mark44 said:
What you've written doesn't make sense to me.
The "tags" of a partition are merely the set of points ##c_i## in the various subintervals.
Definetely true
Mark44 said:
When you say "thus integrability," that's trivially true since the assumption is that "if f is integrable on [a, b]..." This is like saying "if x = 2 <whole bunch of other equations> then x = 2."
Yeah, runnig in circles, it's the raw evidence
andrewkirk said:
You didn't define the lower and upper Darboux sums ##L(f,P), U(f,P)## but let's use the standard definitions from here. The relationship follows automatically from the definition since ##t_i\in[x_i,x_{i+1}]## gives us:
$$\inf_{t\in [x_i,x_{i+1}]} f(t) \le f(t_i) \le \sup_{t\in [x_i,x_{i+1}]} f(t)$$
We then sum the inequality over ##i## to get the result.
Is there an equivalence relationship between partition tags and bounded Riemann sums, as claimed in the original post? Quoted above is the way to prove it, if so? If that is the case, could I be given the next step?
Regards!
 
  • #9
Hi, PF
andrewkirk said:
You didn't define the lower and upper Darboux sums ##L(f,P), U(f,P)## but let's use the standard definitions from here. The relationship follows automatically from the definition since ##t_i\in[x_i,x_{i+1}]## gives us:
$$\inf_{t\in [x_i,x_{i+1}]} f(t) \le f(t_i) \le \sup_{t\in [x_i,x_{i+1}]} f(t)$$
We then sum the inequality over ##i## to get the result.
$$\underset{t\in[x_i,x_{i+1}]}{\inf{f(t)}}\leq f(t_i)\leq\underset{t\in[x_i,x_{i+1}]}{\sup{f(t)}}$$

$$\displaystyle\sum_{t=0}^n{\inf f(t)}\leq\displaystyle\sum_{i=0}^n{f(t_i)}\leq\displaystyle\sum_{t=0}^n{\sup f(t)}$$

$$L(f)=\displaystyle\lim_{t_i \to\infty}{\displaystyle\sum_{i=0}^{\infty}{\inf f(t_i)}}\leq\displaystyle\lim_{t_i \to\infty}{f(t_i)}\leq \displaystyle\lim_{t_i \to\infty}{\displaystyle\sum_{i=0}^{\infty}{\sup f(t_i)}}=U(f)$$

##\therefore## ##f(t)## is Darboux integrable, where

$$L(f)=\inf\{L(f,P)|P\in{\mathcal P([x_i,x_{i+1}])}\}$$
$$U(f)=\sup\{U(f,P)|P\in{\mathcal P([x_i,x_{i+1}])}\}$$
Right?
 
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  • #10
Not right 😒
 
  • #11
mcastillo356 said:
Hi, PF

$$\underset{t\in[x_i,x_{i+1}]}{\inf{f(t)}}\leq f(t_i)\leq\underset{t\in[x_i,x_{i+1}]}{\sup{f(t)}}$$

$$\displaystyle\sum_{t=0}^n{\inf f(t)}\leq\displaystyle\sum_{i=0}^n{f(t_i)}\leq\displaystyle\sum_{t=0}^n{\sup f(t)}$$
Yes, not right. As I said in post #2, you need to sum the first inequality over ##i##. Yet you have summed the three parts differently - over ##i## for the middle part but over ##t## in the two outside parts. Why? You need to sum all parts identically for the inequality to continue to hold.
And the range over which the inf and sup are taken (## t\in [x_i,x_{i+1}]##) has disappeared. Why?
 
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  • #12
andrewkirk said:
Yes, not right. As I said in post #2, you need to sum the first inequality over ##i##. Yet you have summed the three parts differently - over ##i## for the middle part but over ##t## in the two outside parts. Why? You need to sum all parts identically for the inequality to continue to hold.
And the range over which the inf and sup are taken (## t\in [x_i,x_{i+1}]##) has disappeared. Why?

It is also necessary to multiply each term by the width of the corresponding subinterval, [itex]x_{i+1} - x_i[/itex].
 
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  • #13
Hi, PF, let's give another try

$$\inf_{t\in{[x_i,x_{i+1}]}}f(t)\le f(t_i)\le \sup_{t\in{[x_i,x_{i+1}]}}f(t)$$
$$\sum_{t=0}^n \inf_{t\in{[x_i,x_{i+1}]}}f(t)(x_{i+1}-x_i)\le \sum_{t=0}^n f(t)(x_{i+1}-x_i)\le \sum_{t=0}^n \sup_{t\in{[x_i,x_{i+1}]}}f(t)(x_{i+1}-x_i)$$
$$\lim_{||N||\to 0}\left(\sum_{t=0}^n \inf_{t\in{[x_i,x_{i+1}]}}f(t)(x_{i+1}-x_i)\le \sum_{t=0}^n f(t)(x_{i+1}-x_i)\le \sum_{t=0}^n \sup_{t\in{[x_i,x_{i+1}]}}f(t)(x_{i+1}-x_i)\right)$$
$$\Leftrightarrow{L(f)=U(f)}$$
$$\Leftrightarrow{\int_a^b f(t)\,dt}$$
where ##a=x_i##, ##b=x_{i+1}##. BW!
 
  • #14
Hi, PF, wrong again :smile:
 
  • #15
Darboux and Riemann integrability are equivalent: [itex]f[/itex] is Darboux integrable iff it is Riemann integrable, and where the two integrals exist they give the same value. The proof should be given in any decent analysis textbook which covers both integrals, or you can find one here. It requires more than the observation that Riemann sums are bounded by the lower and upper Darboux sums.
 
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  • #16
andrewkirk said:
You didn't define the lower and upper Darboux sums ##L(f,P), U(f,P)## but let's use the standard definitions from here. The relationship follows automatically from the definition since ##t_i\in[x_i,x_{i+1}]## gives us:
$$\inf_{t\in [x_i,x_{i+1}]} f(t) \le f(t_i) \le \sup_{t\in [x_i,x_{i+1}]} f(t)$$
We then sum the inequality over ##i## to get the result.
For each interval ##[x_i,x_{i+1}]## of any partition:
##L(f,P)## approximates the area in that section under the function by means of a little rectangle of height the lowest value ##m_i## that the function takes in the interval.
##R(f,P,c)## approximates the area in that section under the function by means of a little rectangle of height the value ##f(c_i)## of the function.
##U(f,P)## approximates the area in that section under the function by means of a little rectangle of height the higher value ##M_i## that the function takes in the interval.
From the fact that ##m_i\le f(c_i)\le M_i##, inmediately it follows
$$L(f,P)\le R(f,P,c)\le U(f,P)$$
If we want to detail more, simply observe that
$$L(f,P)=\sum_{i=0}^{n-1}(x_{i+1}-x_i)m_i\le \sum_{i=0}^{n-1}(x_{i+1}-x_i)f(c_i)=R(f,P,c)\le \sum_{i=0}^{n-1}(x_{i+1}-x_i)M_i=U(f,P)$$
BW, L&P.
 
  • #17
@mcastillo356 Very good. You have proven that the Riemann sum is bounded below and above by the lower and upper Darboux sums.
Is that as far as you wanted to go or do you want to prove additional results?
 

1. What does it mean to express a limit as a definite integral?

Expressing a limit as a definite integral means finding the area under a curve between two specific points on the x-axis, as the x-values approach a certain value. This allows us to calculate the exact value of the limit.

2. When should a limit be expressed as a definite integral?

A limit should be expressed as a definite integral when the function is continuous, and the limit can be evaluated using the Fundamental Theorem of Calculus. This is often the case for functions that represent physical quantities, such as velocity or displacement.

3. What is the process for expressing a limit as a definite integral?

The process involves first rewriting the limit in terms of the variable of integration, and then using the limit definition of a definite integral to evaluate the area under the curve. This involves taking the limit of a Riemann sum as the number of partitions approaches infinity.

4. How does expressing a limit as a definite integral differ from evaluating a limit algebraically?

Expressing a limit as a definite integral allows us to find the exact value of the limit, whereas evaluating a limit algebraically may only give us an approximation. Additionally, using a definite integral allows us to visualize the limit as the area under a curve, rather than just a numerical value.

5. Are there any special cases or exceptions when expressing a limit as a definite integral?

Yes, there are some cases where a limit cannot be expressed as a definite integral. This may occur when the function is discontinuous or when the limit approaches infinity or negative infinity. In these cases, other methods, such as L'Hôpital's rule, may need to be used to evaluate the limit.

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