Show that $sum_i = 1^n f(i) = lfloor n rfloor f(n)- int_1^nf'(x)lfloor xrfloor, dx$ The 2019 Stack Overflow Developer Survey Results Are InHow to prove this property of floor function?Transforming a Riemann-Stieltjes integral related to the factorial$varepsilon$-$delta$ proof of continuity of floor function $lfloor xrfloor$Solve $lim_xto +inftyfracx^x(lfloor x rfloor)^lfloor x rfloor $Is $f(x)-leftlfloor f(x)rightrfloor$ continuous?Integrating with floor-function:$int_a^bf'(x)lfloor xrfloor dx $Sum the series $sum_n=0^inftya_n$ if $a_n =(-1)^lfloorfracnmrfloor cdot r^n$if $f(x)=int_0^x lfloortrfloor ,dt$ for $x≥0$, draw the graph of $f$ over the interval [0,4]Is the sum of series $sum_n=0^infty lfloor npi rfloor x^n$ a rational function?Prove that $f(x)=x^2 cdot lfloor frac1x^2rfloor$ is continuous.
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Show that $sum_i = 1^n f(i) = lfloor n rfloor f(n)- int_1^nf'(x)lfloor xrfloor, dx$
The 2019 Stack Overflow Developer Survey Results Are InHow to prove this property of floor function?Transforming a Riemann-Stieltjes integral related to the factorial$varepsilon$-$delta$ proof of continuity of floor function $lfloor xrfloor$Solve $lim_xto +inftyfracx^x(lfloor x rfloor)^lfloor x rfloor $Is $f(x)-leftlfloor f(x)rightrfloor$ continuous?Integrating with floor-function:$int_a^bf'(x)lfloor xrfloor dx $Sum the series $sum_n=0^inftya_n$ if $a_n =(-1)^lfloorfracnmrfloor cdot r^n$if $f(x)=int_0^x lfloortrfloor ,dt$ for $x≥0$, draw the graph of $f$ over the interval [0,4]Is the sum of series $sum_n=0^infty lfloor npi rfloor x^n$ a rational function?Prove that $f(x)=x^2 cdot lfloor frac1x^2rfloor$ is continuous.
$begingroup$
Where $f$ is a function defined in $mathbbR$ with countinuos derivative in all
$mathbbR$, for each $nin mathbbN$ and the function $lfloor x rfloor$ is the floor function.
I tried using integration by parts:
I have
$$int_1^n f'(x)lfloor x rfloor dx + int_1^n lfloor x rfloor ' f(x) dx = lfloor n rfloor f(n)-lfloor 1 rfloor f(1)$$
Then $$int_1^nf(x)lfloor x rfloor ' dx + lfloor 1 rfloor f(1) = lfloor x rfloor f(n)- int_1^n lfloor x rfloor f'(x) dx$$
But I have(Is this correct?):
$$int_1^nf'(x) lfloor x rfloor dx= sum_i=1^n lfloor x rfloor int_i^i+1f'(x) dx= sum_i = 1^n f(i)$$
So now I end with:
$$sum_i = 1^n f(i)= lfloor x rfloor f(n) -int_1^nf(x)lfloor x rfloor ' dx - lfloor 1 rfloor f(1) $$
But I don't know how to proceed from here.
I see this problem in the book Real Analysis from Carothers
14.37.a
real-analysis riemann-integration stieltjes-integral
asked Apr 6 at 19:45
RebecaRRebecaR
528
New contributor
RebecaR is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Where $f$ is a function defined in $mathbbR$ with countinuos derivative in all
$mathbbR$, for each $nin mathbbN$ and the function $lfloor x rfloor$ is the floor function.
I tried using integration by parts:
I have
$$int_1^n f'(x)lfloor x rfloor dx + int_1^n lfloor x rfloor ' f(x) dx = lfloor n rfloor f(n)-lfloor 1 rfloor f(1)$$
Then $$int_1^nf(x)lfloor x rfloor ' dx + lfloor 1 rfloor f(1) = lfloor x rfloor f(n)- int_1^n lfloor x rfloor f'(x) dx$$
But I have(Is this correct?):
$$int_1^nf'(x) lfloor x rfloor dx= sum_i=1^n lfloor x rfloor int_i^i+1f'(x) dx= sum_i = 1^n f(i)$$
So now I end with:
$$sum_i = 1^n f(i)= lfloor x rfloor f(n) -int_1^nf(x)lfloor x rfloor ' dx - lfloor 1 rfloor f(1) $$
But I don't know how to proceed from here.
I see this problem in the book Real Analysis from Carothers
14.37.a
real-analysis riemann-integration stieltjes-integral
asked Apr 6 at 19:45
RebecaRRebecaR
528
New contributor
RebecaR is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
I self learning real Analysis, ths is where the book i'm following introduces integrator of bounded variable but i don't understand this yet, so I'm trying to do all the excercises from this section.
$endgroup$
– RebecaR
Apr 6 at 21:52
$begingroup$
en.m.wikipedia.org/wiki/Summation_by_parts
$endgroup$
– HAMIDINE SOUMARE
Apr 6 at 22:12
add a comment |
$begingroup$
Where $f$ is a function defined in $mathbbR$ with countinuos derivative in all
$mathbbR$, for each $nin mathbbN$ and the function $lfloor x rfloor$ is the floor function.
I tried using integration by parts:
I have
$$int_1^n f'(x)lfloor x rfloor dx + int_1^n lfloor x rfloor ' f(x) dx = lfloor n rfloor f(n)-lfloor 1 rfloor f(1)$$
Then $$int_1^nf(x)lfloor x rfloor ' dx + lfloor 1 rfloor f(1) = lfloor x rfloor f(n)- int_1^n lfloor x rfloor f'(x) dx$$
But I have(Is this correct?):
$$int_1^nf'(x) lfloor x rfloor dx= sum_i=1^n lfloor x rfloor int_i^i+1f'(x) dx= sum_i = 1^n f(i)$$
So now I end with:
$$sum_i = 1^n f(i)= lfloor x rfloor f(n) -int_1^nf(x)lfloor x rfloor ' dx - lfloor 1 rfloor f(1) $$
But I don't know how to proceed from here.
I see this problem in the book Real Analysis from Carothers
14.37.a
real-analysis riemann-integration stieltjes-integral
asked Apr 6 at 19:45
RebecaRRebecaR
528
New contributor
RebecaR is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Where $f$ is a function defined in $mathbbR$ with countinuos derivative in all
$mathbbR$, for each $nin mathbbN$ and the function $lfloor x rfloor$ is the floor function.
I tried using integration by parts:
I have
$$int_1^n f'(x)lfloor x rfloor dx + int_1^n lfloor x rfloor ' f(x) dx = lfloor n rfloor f(n)-lfloor 1 rfloor f(1)$$
Then $$int_1^nf(x)lfloor x rfloor ' dx + lfloor 1 rfloor f(1) = lfloor x rfloor f(n)- int_1^n lfloor x rfloor f'(x) dx$$
But I have(Is this correct?):
$$int_1^nf'(x) lfloor x rfloor dx= sum_i=1^n lfloor x rfloor int_i^i+1f'(x) dx= sum_i = 1^n f(i)$$
So now I end with:
$$sum_i = 1^n f(i)= lfloor x rfloor f(n) -int_1^nf(x)lfloor x rfloor ' dx - lfloor 1 rfloor f(1) $$
But I don't know how to proceed from here.
I see this problem in the book Real Analysis from Carothers
14.37.a
real-analysis riemann-integration stieltjes-integral
real-analysis riemann-integration stieltjes-integral
asked Apr 6 at 19:45
RebecaRRebecaR
528
New contributor
RebecaR is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Apr 6 at 19:45
RebecaRRebecaR
528
New contributor
RebecaR is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Apr 7 at 18:24
RebecaR
asked Apr 6 at 19:45
RebecaRRebecaR
528
New contributor
RebecaR is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Apr 6 at 19:45
RebecaRRebecaR
528
asked Apr 6 at 19:45
RebecaRRebecaR
528
528
New contributor
RebecaR is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
RebecaR is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
RebecaR is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
I self learning real Analysis, ths is where the book i'm following introduces integrator of bounded variable but i don't understand this yet, so I'm trying to do all the excercises from this section.
$endgroup$
– RebecaR
Apr 6 at 21:52
$begingroup$
en.m.wikipedia.org/wiki/Summation_by_parts
$endgroup$
– HAMIDINE SOUMARE
Apr 6 at 22:12
add a comment |
$begingroup$
I self learning real Analysis, ths is where the book i'm following introduces integrator of bounded variable but i don't understand this yet, so I'm trying to do all the excercises from this section.
$endgroup$
– RebecaR
Apr 6 at 21:52
$begingroup$
en.m.wikipedia.org/wiki/Summation_by_parts
$endgroup$
– HAMIDINE SOUMARE
Apr 6 at 22:12
$begingroup$
I self learning real Analysis, ths is where the book i'm following introduces integrator of bounded variable but i don't understand this yet, so I'm trying to do all the excercises from this section.
$endgroup$
– RebecaR
Apr 6 at 21:52
$begingroup$
I self learning real Analysis, ths is where the book i'm following introduces integrator of bounded variable but i don't understand this yet, so I'm trying to do all the excercises from this section.
$endgroup$
– RebecaR
Apr 6 at 21:52
$begingroup$
en.m.wikipedia.org/wiki/Summation_by_parts
$endgroup$
– HAMIDINE SOUMARE
Apr 6 at 22:12
$begingroup$
en.m.wikipedia.org/wiki/Summation_by_parts
$endgroup$
– HAMIDINE SOUMARE
Apr 6 at 22:12
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
HINT: note that $lfloor xrfloorle x$, and that $lfloor xrfloor$ is a constant function in $[n,n+1)$ for any chosen $ninBbb N$, so
$$int_1^n f'(x)lfloor xrfloor, dx\=int_1^2 f'(x)cdot 1, dx+int_2^3 f'(x)cdot 2, dx+ldots+int_n-1^n f'(x)(n-1), dx\
=sum_k=1^n-1kint_k^k+1f'(x), dxtag1$$
Now applying the fundamental theorem of calculus we find that
$$sum_k=1^n-1kint_k^k+1f'(x), dx=sum_k=1^n-1k(f(k+1)-f(k))=sum_k=1^n-1 kf(k+1)-sum_k=1^n-1kf(k)\
=sum_k=2^n (k-1)f(k)-sum_k=1^n-1kf(k)=(n-1)f(n)-f(1)-sum_k=2^n-1f(k)$$
Can you conclude from here, right?
answered Apr 6 at 21:35
MasacrosoMasacroso
13.1k41747
$endgroup$
$begingroup$
Sure, you solve it all. Maybe I confuse both becouse i tried to use integration by parts.
$endgroup$
– RebecaR
Apr 6 at 22:40
$begingroup$
@RebecaR I skimmed the book, and probably the exercise want to be solved in a different way (more complicated), anyway Im not completely sure
$endgroup$
– Masacroso
Apr 6 at 22:50
add a comment |
Your Answer
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1 Answer
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active
oldest
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1 Answer
1
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oldest
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active
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active
oldest
votes
$begingroup$
HINT: note that $lfloor xrfloorle x$, and that $lfloor xrfloor$ is a constant function in $[n,n+1)$ for any chosen $ninBbb N$, so
$$int_1^n f'(x)lfloor xrfloor, dx\=int_1^2 f'(x)cdot 1, dx+int_2^3 f'(x)cdot 2, dx+ldots+int_n-1^n f'(x)(n-1), dx\
=sum_k=1^n-1kint_k^k+1f'(x), dxtag1$$
Now applying the fundamental theorem of calculus we find that
$$sum_k=1^n-1kint_k^k+1f'(x), dx=sum_k=1^n-1k(f(k+1)-f(k))=sum_k=1^n-1 kf(k+1)-sum_k=1^n-1kf(k)\
=sum_k=2^n (k-1)f(k)-sum_k=1^n-1kf(k)=(n-1)f(n)-f(1)-sum_k=2^n-1f(k)$$
Can you conclude from here, right?
answered Apr 6 at 21:35
MasacrosoMasacroso
13.1k41747
$endgroup$
$begingroup$
Sure, you solve it all. Maybe I confuse both becouse i tried to use integration by parts.
$endgroup$
– RebecaR
Apr 6 at 22:40
$begingroup$
@RebecaR I skimmed the book, and probably the exercise want to be solved in a different way (more complicated), anyway Im not completely sure
$endgroup$
– Masacroso
Apr 6 at 22:50
add a comment |
$begingroup$
HINT: note that $lfloor xrfloorle x$, and that $lfloor xrfloor$ is a constant function in $[n,n+1)$ for any chosen $ninBbb N$, so
$$int_1^n f'(x)lfloor xrfloor, dx\=int_1^2 f'(x)cdot 1, dx+int_2^3 f'(x)cdot 2, dx+ldots+int_n-1^n f'(x)(n-1), dx\
=sum_k=1^n-1kint_k^k+1f'(x), dxtag1$$
Now applying the fundamental theorem of calculus we find that
$$sum_k=1^n-1kint_k^k+1f'(x), dx=sum_k=1^n-1k(f(k+1)-f(k))=sum_k=1^n-1 kf(k+1)-sum_k=1^n-1kf(k)\
=sum_k=2^n (k-1)f(k)-sum_k=1^n-1kf(k)=(n-1)f(n)-f(1)-sum_k=2^n-1f(k)$$
Can you conclude from here, right?
answered Apr 6 at 21:35
MasacrosoMasacroso
13.1k41747
$endgroup$
$begingroup$
Sure, you solve it all. Maybe I confuse both becouse i tried to use integration by parts.
$endgroup$
– RebecaR
Apr 6 at 22:40
$begingroup$
@RebecaR I skimmed the book, and probably the exercise want to be solved in a different way (more complicated), anyway Im not completely sure
$endgroup$
– Masacroso
Apr 6 at 22:50
add a comment |
$begingroup$
HINT: note that $lfloor xrfloorle x$, and that $lfloor xrfloor$ is a constant function in $[n,n+1)$ for any chosen $ninBbb N$, so
$$int_1^n f'(x)lfloor xrfloor, dx\=int_1^2 f'(x)cdot 1, dx+int_2^3 f'(x)cdot 2, dx+ldots+int_n-1^n f'(x)(n-1), dx\
=sum_k=1^n-1kint_k^k+1f'(x), dxtag1$$
Now applying the fundamental theorem of calculus we find that
$$sum_k=1^n-1kint_k^k+1f'(x), dx=sum_k=1^n-1k(f(k+1)-f(k))=sum_k=1^n-1 kf(k+1)-sum_k=1^n-1kf(k)\
=sum_k=2^n (k-1)f(k)-sum_k=1^n-1kf(k)=(n-1)f(n)-f(1)-sum_k=2^n-1f(k)$$
Can you conclude from here, right?
answered Apr 6 at 21:35
MasacrosoMasacroso
13.1k41747
$endgroup$
HINT: note that $lfloor xrfloorle x$, and that $lfloor xrfloor$ is a constant function in $[n,n+1)$ for any chosen $ninBbb N$, so
$$int_1^n f'(x)lfloor xrfloor, dx\=int_1^2 f'(x)cdot 1, dx+int_2^3 f'(x)cdot 2, dx+ldots+int_n-1^n f'(x)(n-1), dx\
=sum_k=1^n-1kint_k^k+1f'(x), dxtag1$$
Now applying the fundamental theorem of calculus we find that
$$sum_k=1^n-1kint_k^k+1f'(x), dx=sum_k=1^n-1k(f(k+1)-f(k))=sum_k=1^n-1 kf(k+1)-sum_k=1^n-1kf(k)\
=sum_k=2^n (k-1)f(k)-sum_k=1^n-1kf(k)=(n-1)f(n)-f(1)-sum_k=2^n-1f(k)$$
Can you conclude from here, right?
answered Apr 6 at 21:35
MasacrosoMasacroso
13.1k41747
edited Apr 6 at 22:15
answered Apr 6 at 21:35
MasacrosoMasacroso
13.1k41747
answered Apr 6 at 21:35
MasacrosoMasacroso
13.1k41747
answered Apr 6 at 21:35
MasacrosoMasacroso
13.1k41747
13.1k41747
$begingroup$
Sure, you solve it all. Maybe I confuse both becouse i tried to use integration by parts.
$endgroup$
– RebecaR
Apr 6 at 22:40
$begingroup$
@RebecaR I skimmed the book, and probably the exercise want to be solved in a different way (more complicated), anyway Im not completely sure
$endgroup$
– Masacroso
Apr 6 at 22:50
add a comment |
$begingroup$
Sure, you solve it all. Maybe I confuse both becouse i tried to use integration by parts.
$endgroup$
– RebecaR
Apr 6 at 22:40
$begingroup$
@RebecaR I skimmed the book, and probably the exercise want to be solved in a different way (more complicated), anyway Im not completely sure
$endgroup$
– Masacroso
Apr 6 at 22:50
$begingroup$
Sure, you solve it all. Maybe I confuse both becouse i tried to use integration by parts.
$endgroup$
– RebecaR
Apr 6 at 22:40
$begingroup$
Sure, you solve it all. Maybe I confuse both becouse i tried to use integration by parts.
$endgroup$
– RebecaR
Apr 6 at 22:40
$begingroup$
@RebecaR I skimmed the book, and probably the exercise want to be solved in a different way (more complicated), anyway Im not completely sure
$endgroup$
– Masacroso
Apr 6 at 22:50
$begingroup$
@RebecaR I skimmed the book, and probably the exercise want to be solved in a different way (more complicated), anyway Im not completely sure
$endgroup$
– Masacroso
Apr 6 at 22:50
add a comment |
RebecaR is a new contributor. Be nice, and check out our Code of Conduct.
RebecaR is a new contributor. Be nice, and check out our Code of Conduct.
RebecaR is a new contributor. Be nice, and check out our Code of Conduct.
RebecaR is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
I self learning real Analysis, ths is where the book i'm following introduces integrator of bounded variable but i don't understand this yet, so I'm trying to do all the excercises from this section.
$endgroup$
– RebecaR
Apr 6 at 21:52
$begingroup$
en.m.wikipedia.org/wiki/Summation_by_parts
$endgroup$
– HAMIDINE SOUMARE
Apr 6 at 22:12