Sum of reciprocals of numbers with certain terms omitted












32












$begingroup$


I know that the harmonic series $1 + frac12 + frac13 + frac14 + cdots$ diverges. I also know that the sum of the inverse of prime numbers $frac12 + frac13 + frac15 + frac17 + frac1{11} + cdots$ diverges too, even if really slowly since it's $O(log log n)$.



But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite?






sequences-and-series convergence







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    You might want to rephrase your question as "Sum of Harmonic Series with certain terms omitted". It might be easier to immediately understand what you mean =) But +1 for me; this is a good question that this site needs.
    $endgroup$
    – Justin L.
    Jul 21 '10 at 21:03












  • $begingroup$
    What a nice question. I would be really interested in the exact values of the Kempner Series...
    $endgroup$
    – Max Muller
    Apr 5 '11 at 15:12










  • $begingroup$
    @mau : A related (but different) question is this one.
    $endgroup$
    – Watson
    Nov 10 '16 at 19:57






  • 1




    $begingroup$
    @Watson: really nice!
    $endgroup$
    – mau
    Nov 10 '16 at 20:21
















32












$begingroup$


I know that the harmonic series $1 + frac12 + frac13 + frac14 + cdots$ diverges. I also know that the sum of the inverse of prime numbers $frac12 + frac13 + frac15 + frac17 + frac1{11} + cdots$ diverges too, even if really slowly since it's $O(log log n)$.



But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite?






sequences-and-series convergence







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    You might want to rephrase your question as "Sum of Harmonic Series with certain terms omitted". It might be easier to immediately understand what you mean =) But +1 for me; this is a good question that this site needs.
    $endgroup$
    – Justin L.
    Jul 21 '10 at 21:03












  • $begingroup$
    What a nice question. I would be really interested in the exact values of the Kempner Series...
    $endgroup$
    – Max Muller
    Apr 5 '11 at 15:12










  • $begingroup$
    @mau : A related (but different) question is this one.
    $endgroup$
    – Watson
    Nov 10 '16 at 19:57






  • 1




    $begingroup$
    @Watson: really nice!
    $endgroup$
    – mau
    Nov 10 '16 at 20:21














32












32








32


17



$begingroup$


I know that the harmonic series $1 + frac12 + frac13 + frac14 + cdots$ diverges. I also know that the sum of the inverse of prime numbers $frac12 + frac13 + frac15 + frac17 + frac1{11} + cdots$ diverges too, even if really slowly since it's $O(log log n)$.



But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite?






sequences-and-series convergence







share|cite|improve this question











$endgroup$




I know that the harmonic series $1 + frac12 + frac13 + frac14 + cdots$ diverges. I also know that the sum of the inverse of prime numbers $frac12 + frac13 + frac15 + frac17 + frac1{11} + cdots$ diverges too, even if really slowly since it's $O(log log n)$.



But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite?






sequences-and-series convergence




sequences-and-series convergence






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jul 9 '15 at 20:54









Olivier Oloa

108k17177294




108k17177294










asked Jul 21 '10 at 20:57









maumau

7,07523263




7,07523263








  • 3




    $begingroup$
    You might want to rephrase your question as "Sum of Harmonic Series with certain terms omitted". It might be easier to immediately understand what you mean =) But +1 for me; this is a good question that this site needs.
    $endgroup$
    – Justin L.
    Jul 21 '10 at 21:03












  • $begingroup$
    What a nice question. I would be really interested in the exact values of the Kempner Series...
    $endgroup$
    – Max Muller
    Apr 5 '11 at 15:12










  • $begingroup$
    @mau : A related (but different) question is this one.
    $endgroup$
    – Watson
    Nov 10 '16 at 19:57






  • 1




    $begingroup$
    @Watson: really nice!
    $endgroup$
    – mau
    Nov 10 '16 at 20:21














  • 3




    $begingroup$
    You might want to rephrase your question as "Sum of Harmonic Series with certain terms omitted". It might be easier to immediately understand what you mean =) But +1 for me; this is a good question that this site needs.
    $endgroup$
    – Justin L.
    Jul 21 '10 at 21:03












  • $begingroup$
    What a nice question. I would be really interested in the exact values of the Kempner Series...
    $endgroup$
    – Max Muller
    Apr 5 '11 at 15:12










  • $begingroup$
    @mau : A related (but different) question is this one.
    $endgroup$
    – Watson
    Nov 10 '16 at 19:57






  • 1




    $begingroup$
    @Watson: really nice!
    $endgroup$
    – mau
    Nov 10 '16 at 20:21








3




3




$begingroup$
You might want to rephrase your question as "Sum of Harmonic Series with certain terms omitted". It might be easier to immediately understand what you mean =) But +1 for me; this is a good question that this site needs.
$endgroup$
– Justin L.
Jul 21 '10 at 21:03






$begingroup$
You might want to rephrase your question as "Sum of Harmonic Series with certain terms omitted". It might be easier to immediately understand what you mean =) But +1 for me; this is a good question that this site needs.
$endgroup$
– Justin L.
Jul 21 '10 at 21:03














$begingroup$
What a nice question. I would be really interested in the exact values of the Kempner Series...
$endgroup$
– Max Muller
Apr 5 '11 at 15:12




$begingroup$
What a nice question. I would be really interested in the exact values of the Kempner Series...
$endgroup$
– Max Muller
Apr 5 '11 at 15:12












$begingroup$
@mau : A related (but different) question is this one.
$endgroup$
– Watson
Nov 10 '16 at 19:57




$begingroup$
@mau : A related (but different) question is this one.
$endgroup$
– Watson
Nov 10 '16 at 19:57




1




1




$begingroup$
@Watson: really nice!
$endgroup$
– mau
Nov 10 '16 at 20:21




$begingroup$
@Watson: really nice!
$endgroup$
– mau
Nov 10 '16 at 20:21










2 Answers
2






active

oldest

votes


















19












$begingroup$

EDIT: This might be what you're looking for. Found it from looking at the source below. They're called Kempner series.



An article here (and cited below) says that one Dr. Kempner proved in 1914 that the series 1+ 1/2 + 1/3 + ..., with any term that has a 9 in the denominator removed, is convergent (though he doesn't say what it converges to in the introductory paragraph). The article goes on to generalize the result.



A Curious Convergent Series
Frank Irwin
The American Mathematical Monthly, Vol. 23, No. 5 (May, 1916), pp. 149-152
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2974352






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thank you - my memory was a bit fallacious on the actual sums, and totally fallacious with respect to the name ok Kempner :-)
    $endgroup$
    – mau
    Jul 21 '10 at 21:12










  • $begingroup$
    Wow, this is a very interesting result
    $endgroup$
    – Casebash
    Jul 21 '10 at 21:12



















39












$begingroup$

It is not very surprising that the sum is finite, since numbers without a 7 (or any other digit) get rarer and rarer as the number of digits increases.



Here's a proof.



Let $S$ be the harmonic series with all terms whose denominator contains the digit $k$ removed. We can write $S =S_1 + S_2 + S_3 + ldots$, where $S_i$ is the sum of all terms whose denominator contains exactly $i$ digits, all different from $k$.



Now, the number of $i$-digit numbers that do not contain the digit $k$ is $8cdot9^{i-1}$ (there are $8$ choices for the first digit, excluding $0$ and $k$, and $9$ choices for the other digits). [Well, if $k=0$ there are $9$ choices for the first digit, but the proof still works.] So there are $8cdot9^{i-1}$ numbers in the sum $S_i$.



Now each number in $S_i$ is of the form $frac1a$, where $a$ is an $i$-digit number. So $a geq 10^{i-1}$, which implies $frac1a leq frac1{10^{i-1}}$.



Therefore $S_i leq 8cdotdfrac{9^{i-1} }{10^{i-1}} = 8cdotleft(frac9{10}right)^{i-1}$.



So $S= sum S_i leq sum 8cdotleft(frac9{10}right)^{i-1}$



which is a geometric series of ratio $frac9{10} < 1$, which converges. Since $S$ is a positive series bounded above by a converging series, $S$ converges.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    +1: Great Answer!
    $endgroup$
    – Daniel Allen Langdon
    Dec 23 '11 at 16:00











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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









19












$begingroup$

EDIT: This might be what you're looking for. Found it from looking at the source below. They're called Kempner series.



An article here (and cited below) says that one Dr. Kempner proved in 1914 that the series 1+ 1/2 + 1/3 + ..., with any term that has a 9 in the denominator removed, is convergent (though he doesn't say what it converges to in the introductory paragraph). The article goes on to generalize the result.



A Curious Convergent Series
Frank Irwin
The American Mathematical Monthly, Vol. 23, No. 5 (May, 1916), pp. 149-152
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2974352






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thank you - my memory was a bit fallacious on the actual sums, and totally fallacious with respect to the name ok Kempner :-)
    $endgroup$
    – mau
    Jul 21 '10 at 21:12










  • $begingroup$
    Wow, this is a very interesting result
    $endgroup$
    – Casebash
    Jul 21 '10 at 21:12
















19












$begingroup$

EDIT: This might be what you're looking for. Found it from looking at the source below. They're called Kempner series.



An article here (and cited below) says that one Dr. Kempner proved in 1914 that the series 1+ 1/2 + 1/3 + ..., with any term that has a 9 in the denominator removed, is convergent (though he doesn't say what it converges to in the introductory paragraph). The article goes on to generalize the result.



A Curious Convergent Series
Frank Irwin
The American Mathematical Monthly, Vol. 23, No. 5 (May, 1916), pp. 149-152
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2974352






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thank you - my memory was a bit fallacious on the actual sums, and totally fallacious with respect to the name ok Kempner :-)
    $endgroup$
    – mau
    Jul 21 '10 at 21:12










  • $begingroup$
    Wow, this is a very interesting result
    $endgroup$
    – Casebash
    Jul 21 '10 at 21:12














19












19








19





$begingroup$

EDIT: This might be what you're looking for. Found it from looking at the source below. They're called Kempner series.



An article here (and cited below) says that one Dr. Kempner proved in 1914 that the series 1+ 1/2 + 1/3 + ..., with any term that has a 9 in the denominator removed, is convergent (though he doesn't say what it converges to in the introductory paragraph). The article goes on to generalize the result.



A Curious Convergent Series
Frank Irwin
The American Mathematical Monthly, Vol. 23, No. 5 (May, 1916), pp. 149-152
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2974352






share|cite|improve this answer









$endgroup$



EDIT: This might be what you're looking for. Found it from looking at the source below. They're called Kempner series.



An article here (and cited below) says that one Dr. Kempner proved in 1914 that the series 1+ 1/2 + 1/3 + ..., with any term that has a 9 in the denominator removed, is convergent (though he doesn't say what it converges to in the introductory paragraph). The article goes on to generalize the result.



A Curious Convergent Series
Frank Irwin
The American Mathematical Monthly, Vol. 23, No. 5 (May, 1916), pp. 149-152
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2974352







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jul 21 '10 at 21:06









Jan GorznyJan Gorzny

792914




792914












  • $begingroup$
    thank you - my memory was a bit fallacious on the actual sums, and totally fallacious with respect to the name ok Kempner :-)
    $endgroup$
    – mau
    Jul 21 '10 at 21:12










  • $begingroup$
    Wow, this is a very interesting result
    $endgroup$
    – Casebash
    Jul 21 '10 at 21:12


















  • $begingroup$
    thank you - my memory was a bit fallacious on the actual sums, and totally fallacious with respect to the name ok Kempner :-)
    $endgroup$
    – mau
    Jul 21 '10 at 21:12










  • $begingroup$
    Wow, this is a very interesting result
    $endgroup$
    – Casebash
    Jul 21 '10 at 21:12
















$begingroup$
thank you - my memory was a bit fallacious on the actual sums, and totally fallacious with respect to the name ok Kempner :-)
$endgroup$
– mau
Jul 21 '10 at 21:12




$begingroup$
thank you - my memory was a bit fallacious on the actual sums, and totally fallacious with respect to the name ok Kempner :-)
$endgroup$
– mau
Jul 21 '10 at 21:12












$begingroup$
Wow, this is a very interesting result
$endgroup$
– Casebash
Jul 21 '10 at 21:12




$begingroup$
Wow, this is a very interesting result
$endgroup$
– Casebash
Jul 21 '10 at 21:12











39












$begingroup$

It is not very surprising that the sum is finite, since numbers without a 7 (or any other digit) get rarer and rarer as the number of digits increases.



Here's a proof.



Let $S$ be the harmonic series with all terms whose denominator contains the digit $k$ removed. We can write $S =S_1 + S_2 + S_3 + ldots$, where $S_i$ is the sum of all terms whose denominator contains exactly $i$ digits, all different from $k$.



Now, the number of $i$-digit numbers that do not contain the digit $k$ is $8cdot9^{i-1}$ (there are $8$ choices for the first digit, excluding $0$ and $k$, and $9$ choices for the other digits). [Well, if $k=0$ there are $9$ choices for the first digit, but the proof still works.] So there are $8cdot9^{i-1}$ numbers in the sum $S_i$.



Now each number in $S_i$ is of the form $frac1a$, where $a$ is an $i$-digit number. So $a geq 10^{i-1}$, which implies $frac1a leq frac1{10^{i-1}}$.



Therefore $S_i leq 8cdotdfrac{9^{i-1} }{10^{i-1}} = 8cdotleft(frac9{10}right)^{i-1}$.



So $S= sum S_i leq sum 8cdotleft(frac9{10}right)^{i-1}$



which is a geometric series of ratio $frac9{10} < 1$, which converges. Since $S$ is a positive series bounded above by a converging series, $S$ converges.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    +1: Great Answer!
    $endgroup$
    – Daniel Allen Langdon
    Dec 23 '11 at 16:00
















39












$begingroup$

It is not very surprising that the sum is finite, since numbers without a 7 (or any other digit) get rarer and rarer as the number of digits increases.



Here's a proof.



Let $S$ be the harmonic series with all terms whose denominator contains the digit $k$ removed. We can write $S =S_1 + S_2 + S_3 + ldots$, where $S_i$ is the sum of all terms whose denominator contains exactly $i$ digits, all different from $k$.



Now, the number of $i$-digit numbers that do not contain the digit $k$ is $8cdot9^{i-1}$ (there are $8$ choices for the first digit, excluding $0$ and $k$, and $9$ choices for the other digits). [Well, if $k=0$ there are $9$ choices for the first digit, but the proof still works.] So there are $8cdot9^{i-1}$ numbers in the sum $S_i$.



Now each number in $S_i$ is of the form $frac1a$, where $a$ is an $i$-digit number. So $a geq 10^{i-1}$, which implies $frac1a leq frac1{10^{i-1}}$.



Therefore $S_i leq 8cdotdfrac{9^{i-1} }{10^{i-1}} = 8cdotleft(frac9{10}right)^{i-1}$.



So $S= sum S_i leq sum 8cdotleft(frac9{10}right)^{i-1}$



which is a geometric series of ratio $frac9{10} < 1$, which converges. Since $S$ is a positive series bounded above by a converging series, $S$ converges.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    +1: Great Answer!
    $endgroup$
    – Daniel Allen Langdon
    Dec 23 '11 at 16:00














39












39








39





$begingroup$

It is not very surprising that the sum is finite, since numbers without a 7 (or any other digit) get rarer and rarer as the number of digits increases.



Here's a proof.



Let $S$ be the harmonic series with all terms whose denominator contains the digit $k$ removed. We can write $S =S_1 + S_2 + S_3 + ldots$, where $S_i$ is the sum of all terms whose denominator contains exactly $i$ digits, all different from $k$.



Now, the number of $i$-digit numbers that do not contain the digit $k$ is $8cdot9^{i-1}$ (there are $8$ choices for the first digit, excluding $0$ and $k$, and $9$ choices for the other digits). [Well, if $k=0$ there are $9$ choices for the first digit, but the proof still works.] So there are $8cdot9^{i-1}$ numbers in the sum $S_i$.



Now each number in $S_i$ is of the form $frac1a$, where $a$ is an $i$-digit number. So $a geq 10^{i-1}$, which implies $frac1a leq frac1{10^{i-1}}$.



Therefore $S_i leq 8cdotdfrac{9^{i-1} }{10^{i-1}} = 8cdotleft(frac9{10}right)^{i-1}$.



So $S= sum S_i leq sum 8cdotleft(frac9{10}right)^{i-1}$



which is a geometric series of ratio $frac9{10} < 1$, which converges. Since $S$ is a positive series bounded above by a converging series, $S$ converges.






share|cite|improve this answer











$endgroup$



It is not very surprising that the sum is finite, since numbers without a 7 (or any other digit) get rarer and rarer as the number of digits increases.



Here's a proof.



Let $S$ be the harmonic series with all terms whose denominator contains the digit $k$ removed. We can write $S =S_1 + S_2 + S_3 + ldots$, where $S_i$ is the sum of all terms whose denominator contains exactly $i$ digits, all different from $k$.



Now, the number of $i$-digit numbers that do not contain the digit $k$ is $8cdot9^{i-1}$ (there are $8$ choices for the first digit, excluding $0$ and $k$, and $9$ choices for the other digits). [Well, if $k=0$ there are $9$ choices for the first digit, but the proof still works.] So there are $8cdot9^{i-1}$ numbers in the sum $S_i$.



Now each number in $S_i$ is of the form $frac1a$, where $a$ is an $i$-digit number. So $a geq 10^{i-1}$, which implies $frac1a leq frac1{10^{i-1}}$.



Therefore $S_i leq 8cdotdfrac{9^{i-1} }{10^{i-1}} = 8cdotleft(frac9{10}right)^{i-1}$.



So $S= sum S_i leq sum 8cdotleft(frac9{10}right)^{i-1}$



which is a geometric series of ratio $frac9{10} < 1$, which converges. Since $S$ is a positive series bounded above by a converging series, $S$ converges.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Mar 13 '13 at 22:30









P..

13.5k22348




13.5k22348










answered Jul 21 '10 at 21:35









user115user115

1,0451813




1,0451813












  • $begingroup$
    +1: Great Answer!
    $endgroup$
    – Daniel Allen Langdon
    Dec 23 '11 at 16:00


















  • $begingroup$
    +1: Great Answer!
    $endgroup$
    – Daniel Allen Langdon
    Dec 23 '11 at 16:00
















$begingroup$
+1: Great Answer!
$endgroup$
– Daniel Allen Langdon
Dec 23 '11 at 16:00




$begingroup$
+1: Great Answer!
$endgroup$
– Daniel Allen Langdon
Dec 23 '11 at 16:00


















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Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

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up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
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