Let $A={sum_{i=1}^{infty} frac{a_i}{5^{i}}:a_i=0,1,2,3$ or $4 } subset mathbb{R}$. Then which of the...
$begingroup$
Let $$A=bigg{sum_{i=1}^{infty} frac{a_i}{5^{i}} : a_iin{0,1,2,3,4} bigg} subset mathbb{R}.$$ Then which of the following are true:
a. $A$ is a finite set.
b. $A$ is countably infinite.
c. $A$ is uncountable but does not contain an open interval.
d. $A$ contains an open interval.
Each such series is convergent. I could also prove that $A$ is uncountable. I am not able to prove or disprove d.
Thanks for the help!!
real-analysis sequences-and-series convergence
edited Dec 14 '15 at 12:12
Surb
37.4k94375
asked Dec 14 '15 at 11:51
tattwamasi amrutamtattwamasi amrutam
8,19821643
$endgroup$
add a comment |
$begingroup$
Let $$A=bigg{sum_{i=1}^{infty} frac{a_i}{5^{i}} : a_iin{0,1,2,3,4} bigg} subset mathbb{R}.$$ Then which of the following are true:
a. $A$ is a finite set.
b. $A$ is countably infinite.
c. $A$ is uncountable but does not contain an open interval.
d. $A$ contains an open interval.
Each such series is convergent. I could also prove that $A$ is uncountable. I am not able to prove or disprove d.
Thanks for the help!!
real-analysis sequences-and-series convergence
edited Dec 14 '15 at 12:12
Surb
37.4k94375
asked Dec 14 '15 at 11:51
tattwamasi amrutamtattwamasi amrutam
8,19821643
$endgroup$
8
$begingroup$
These are just the real numbers in $[0,1]$ written in base $5$.
$endgroup$
– lulu
Dec 14 '15 at 11:55
$begingroup$
I am a bit confused. Is it like $a_1=0$ , $a_2=1$ etc?
$endgroup$
– Kushal Bhuyan
Dec 14 '15 at 12:18
$begingroup$
@Quintic $a_i$ can be anything
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 4:51
$begingroup$
But you wrote $a_i in {0,1,2,3,4}$
$endgroup$
– Kushal Bhuyan
Dec 16 '15 at 5:00
$begingroup$
@Quintic I mean it is not necessary for $a_i$ to be all the same. That is $a_i$ and $a_j$ can be different for $ine j$
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 5:22
add a comment |
$begingroup$
Let $$A=bigg{sum_{i=1}^{infty} frac{a_i}{5^{i}} : a_iin{0,1,2,3,4} bigg} subset mathbb{R}.$$ Then which of the following are true:
a. $A$ is a finite set.
b. $A$ is countably infinite.
c. $A$ is uncountable but does not contain an open interval.
d. $A$ contains an open interval.
Each such series is convergent. I could also prove that $A$ is uncountable. I am not able to prove or disprove d.
Thanks for the help!!
real-analysis sequences-and-series convergence
edited Dec 14 '15 at 12:12
Surb
37.4k94375
asked Dec 14 '15 at 11:51
tattwamasi amrutamtattwamasi amrutam
8,19821643
$endgroup$
Let $$A=bigg{sum_{i=1}^{infty} frac{a_i}{5^{i}} : a_iin{0,1,2,3,4} bigg} subset mathbb{R}.$$ Then which of the following are true:
a. $A$ is a finite set.
b. $A$ is countably infinite.
c. $A$ is uncountable but does not contain an open interval.
d. $A$ contains an open interval.
Each such series is convergent. I could also prove that $A$ is uncountable. I am not able to prove or disprove d.
Thanks for the help!!
real-analysis sequences-and-series convergence
real-analysis sequences-and-series convergence
edited Dec 14 '15 at 12:12
Surb
37.4k94375
asked Dec 14 '15 at 11:51
tattwamasi amrutamtattwamasi amrutam
8,19821643
edited Dec 14 '15 at 12:12
Surb
37.4k94375
asked Dec 14 '15 at 11:51
tattwamasi amrutamtattwamasi amrutam
8,19821643
edited Dec 14 '15 at 12:12
Surb
37.4k94375
edited Dec 14 '15 at 12:12
Surb
37.4k94375
edited Dec 14 '15 at 12:12
Surb
37.4k94375
37.4k94375
asked Dec 14 '15 at 11:51
tattwamasi amrutamtattwamasi amrutam
8,19821643
asked Dec 14 '15 at 11:51
tattwamasi amrutamtattwamasi amrutam
8,19821643
asked Dec 14 '15 at 11:51
tattwamasi amrutamtattwamasi amrutam
8,19821643
8,19821643
8
$begingroup$
These are just the real numbers in $[0,1]$ written in base $5$.
$endgroup$
– lulu
Dec 14 '15 at 11:55
$begingroup$
I am a bit confused. Is it like $a_1=0$ , $a_2=1$ etc?
$endgroup$
– Kushal Bhuyan
Dec 14 '15 at 12:18
$begingroup$
@Quintic $a_i$ can be anything
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 4:51
$begingroup$
But you wrote $a_i in {0,1,2,3,4}$
$endgroup$
– Kushal Bhuyan
Dec 16 '15 at 5:00
$begingroup$
@Quintic I mean it is not necessary for $a_i$ to be all the same. That is $a_i$ and $a_j$ can be different for $ine j$
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 5:22
add a comment |
8
$begingroup$
These are just the real numbers in $[0,1]$ written in base $5$.
$endgroup$
– lulu
Dec 14 '15 at 11:55
$begingroup$
I am a bit confused. Is it like $a_1=0$ , $a_2=1$ etc?
$endgroup$
– Kushal Bhuyan
Dec 14 '15 at 12:18
$begingroup$
@Quintic $a_i$ can be anything
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 4:51
$begingroup$
But you wrote $a_i in {0,1,2,3,4}$
$endgroup$
– Kushal Bhuyan
Dec 16 '15 at 5:00
$begingroup$
@Quintic I mean it is not necessary for $a_i$ to be all the same. That is $a_i$ and $a_j$ can be different for $ine j$
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 5:22
8
8
$begingroup$
These are just the real numbers in $[0,1]$ written in base $5$.
$endgroup$
– lulu
Dec 14 '15 at 11:55
$begingroup$
These are just the real numbers in $[0,1]$ written in base $5$.
$endgroup$
– lulu
Dec 14 '15 at 11:55
$begingroup$
I am a bit confused. Is it like $a_1=0$ , $a_2=1$ etc?
$endgroup$
– Kushal Bhuyan
Dec 14 '15 at 12:18
$begingroup$
I am a bit confused. Is it like $a_1=0$ , $a_2=1$ etc?
$endgroup$
– Kushal Bhuyan
Dec 14 '15 at 12:18
$begingroup$
@Quintic $a_i$ can be anything
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 4:51
$begingroup$
@Quintic $a_i$ can be anything
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 4:51
$begingroup$
But you wrote $a_i in {0,1,2,3,4}$
$endgroup$
– Kushal Bhuyan
Dec 16 '15 at 5:00
$begingroup$
But you wrote $a_i in {0,1,2,3,4}$
$endgroup$
– Kushal Bhuyan
Dec 16 '15 at 5:00
$begingroup$
@Quintic I mean it is not necessary for $a_i$ to be all the same. That is $a_i$ and $a_j$ can be different for $ine j$
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 5:22
$begingroup$
@Quintic I mean it is not necessary for $a_i$ to be all the same. That is $a_i$ and $a_j$ can be different for $ine j$
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 5:22
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: The elements of A amount to all numbers on the interval [0,1] where the numbers are expressed in base 5 instead of the usual base 10. Do you see it?
answered Dec 14 '15 at 12:06
Patrick LincolnPatrick Lincoln
462
$endgroup$
add a comment |
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1 Answer
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$begingroup$
Hint: The elements of A amount to all numbers on the interval [0,1] where the numbers are expressed in base 5 instead of the usual base 10. Do you see it?
answered Dec 14 '15 at 12:06
Patrick LincolnPatrick Lincoln
462
$endgroup$
add a comment |
$begingroup$
Hint: The elements of A amount to all numbers on the interval [0,1] where the numbers are expressed in base 5 instead of the usual base 10. Do you see it?
answered Dec 14 '15 at 12:06
Patrick LincolnPatrick Lincoln
462
$endgroup$
add a comment |
$begingroup$
Hint: The elements of A amount to all numbers on the interval [0,1] where the numbers are expressed in base 5 instead of the usual base 10. Do you see it?
answered Dec 14 '15 at 12:06
Patrick LincolnPatrick Lincoln
462
$endgroup$
Hint: The elements of A amount to all numbers on the interval [0,1] where the numbers are expressed in base 5 instead of the usual base 10. Do you see it?
answered Dec 14 '15 at 12:06
Patrick LincolnPatrick Lincoln
462
answered Dec 14 '15 at 12:06
Patrick LincolnPatrick Lincoln
462
answered Dec 14 '15 at 12:06
Patrick LincolnPatrick Lincoln
462
answered Dec 14 '15 at 12:06
Patrick LincolnPatrick Lincoln
462
462
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8
$begingroup$
These are just the real numbers in $[0,1]$ written in base $5$.
$endgroup$
– lulu
Dec 14 '15 at 11:55
$begingroup$
I am a bit confused. Is it like $a_1=0$ , $a_2=1$ etc?
$endgroup$
– Kushal Bhuyan
Dec 14 '15 at 12:18
$begingroup$
@Quintic $a_i$ can be anything
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 4:51
$begingroup$
But you wrote $a_i in {0,1,2,3,4}$
$endgroup$
– Kushal Bhuyan
Dec 16 '15 at 5:00
$begingroup$
@Quintic I mean it is not necessary for $a_i$ to be all the same. That is $a_i$ and $a_j$ can be different for $ine j$
$endgroup$
– tattwamasi amrutam
Dec 16 '15 at 5:22