An easy way to solve this limit of a sum?How can I evaluate $sum_n=0^infty(n+1)x^n$?$ sum_k=1^infty lnleft(1 + frac14 k^2right)$ Computing this sumalternating series test of $sum(-1)^nfracsqrtn+1-sqrtnn$Alternative way to solve this limit?evaluate the sum of an alternating harmonic series with a fixed limitShow that a Series DivergesIs there a way to sum up the series give below??Is there an easy way to prove that this series diverges?Evaluating the floor of a tough looking summationUnderstanding summation of infinite series by defining a new functionHow to find the $limlimits_n toinfty frac2^n^2(n!)^2$
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An easy way to solve this limit of a sum?
How can I evaluate $sum_n=0^infty(n+1)x^n$?$ sum_k=1^infty lnleft(1 + frac14 k^2right)$ Computing this sumalternating series test of $sum(-1)^nfracsqrtn+1-sqrtnn$Alternative way to solve this limit?evaluate the sum of an alternating harmonic series with a fixed limitShow that a Series DivergesIs there a way to sum up the series give below??Is there an easy way to prove that this series diverges?Evaluating the floor of a tough looking summationUnderstanding summation of infinite series by defining a new functionHow to find the $limlimits_n toinfty frac2^n^2(n!)^2$
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;
$begingroup$
$$lim _ntoinftysum_k=0^nfrack+110^k$$
What I've tried is to create a function out of it in order to derivate it, but I am getting nowhere. I would like to know if there is an easier way to it (I would like a high school level method for this, if there is one).
sequences-and-series
edited 8 hours ago
Parcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
asked 8 hours ago
Jon9Jon9
364 bronze badges
New contributor
Jon9 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 |
$begingroup$
$$lim _ntoinftysum_k=0^nfrack+110^k$$
What I've tried is to create a function out of it in order to derivate it, but I am getting nowhere. I would like to know if there is an easier way to it (I would like a high school level method for this, if there is one).
sequences-and-series
edited 8 hours ago
Parcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
asked 8 hours ago
Jon9Jon9
364 bronze badges
New contributor
Jon9 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
1
$begingroup$
I'd by splitting it into the sum of $k/10^k$ and $1/10^k$, where the latter is easily computable as it is a geometric series.
$endgroup$
– Viktor Glombik
8 hours ago
3
$begingroup$
Well, if you forget the "high-school level" part (or are willing to be a bit hazy on the justification of the steps) you can always consider the function $f$ defined for $xin(-1,1)$ by $f(x) = sum_k=0^infty x^k+1$. You can compute a closed-form for it and differentiate that. You can also see that $f'(x) = sum_k=0^infty (k+1) x^k$, so what you want if $f'(1/10)$.
$endgroup$
– Clement C.
8 hours ago
1
$begingroup$
Possible duplicate of How can I evaluate $sum_n=0^infty(n+1)x^n$?
$endgroup$
– Martin R
7 hours ago
add a comment |
$begingroup$
$$lim _ntoinftysum_k=0^nfrack+110^k$$
What I've tried is to create a function out of it in order to derivate it, but I am getting nowhere. I would like to know if there is an easier way to it (I would like a high school level method for this, if there is one).
sequences-and-series
edited 8 hours ago
Parcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
asked 8 hours ago
Jon9Jon9
364 bronze badges
New contributor
Jon9 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$$lim _ntoinftysum_k=0^nfrack+110^k$$
What I've tried is to create a function out of it in order to derivate it, but I am getting nowhere. I would like to know if there is an easier way to it (I would like a high school level method for this, if there is one).
sequences-and-series
sequences-and-series
edited 8 hours ago
Parcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
asked 8 hours ago
Jon9Jon9
364 bronze badges
New contributor
Jon9 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 8 hours ago
Parcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
asked 8 hours ago
Jon9Jon9
364 bronze badges
New contributor
Jon9 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 8 hours ago
Parcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
edited 8 hours ago
Parcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
edited 8 hours ago
Parcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
48.2k13 gold badges77 silver badges117 bronze badges
asked 8 hours ago
Jon9Jon9
364 bronze badges
New contributor
Jon9 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
Jon9Jon9
364 bronze badges
asked 8 hours ago
Jon9Jon9
364 bronze badges
364 bronze badges
New contributor
Jon9 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Jon9 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
$begingroup$
I'd by splitting it into the sum of $k/10^k$ and $1/10^k$, where the latter is easily computable as it is a geometric series.
$endgroup$
– Viktor Glombik
8 hours ago
3
$begingroup$
Well, if you forget the "high-school level" part (or are willing to be a bit hazy on the justification of the steps) you can always consider the function $f$ defined for $xin(-1,1)$ by $f(x) = sum_k=0^infty x^k+1$. You can compute a closed-form for it and differentiate that. You can also see that $f'(x) = sum_k=0^infty (k+1) x^k$, so what you want if $f'(1/10)$.
$endgroup$
– Clement C.
8 hours ago
1
$begingroup$
Possible duplicate of How can I evaluate $sum_n=0^infty(n+1)x^n$?
$endgroup$
– Martin R
7 hours ago
add a comment |
1
$begingroup$
I'd by splitting it into the sum of $k/10^k$ and $1/10^k$, where the latter is easily computable as it is a geometric series.
$endgroup$
– Viktor Glombik
8 hours ago
3
$begingroup$
Well, if you forget the "high-school level" part (or are willing to be a bit hazy on the justification of the steps) you can always consider the function $f$ defined for $xin(-1,1)$ by $f(x) = sum_k=0^infty x^k+1$. You can compute a closed-form for it and differentiate that. You can also see that $f'(x) = sum_k=0^infty (k+1) x^k$, so what you want if $f'(1/10)$.
$endgroup$
– Clement C.
8 hours ago
1
$begingroup$
Possible duplicate of How can I evaluate $sum_n=0^infty(n+1)x^n$?
$endgroup$
– Martin R
7 hours ago
1
1
$begingroup$
I'd by splitting it into the sum of $k/10^k$ and $1/10^k$, where the latter is easily computable as it is a geometric series.
$endgroup$
– Viktor Glombik
8 hours ago
$begingroup$
I'd by splitting it into the sum of $k/10^k$ and $1/10^k$, where the latter is easily computable as it is a geometric series.
$endgroup$
– Viktor Glombik
8 hours ago
3
3
$begingroup$
Well, if you forget the "high-school level" part (or are willing to be a bit hazy on the justification of the steps) you can always consider the function $f$ defined for $xin(-1,1)$ by $f(x) = sum_k=0^infty x^k+1$. You can compute a closed-form for it and differentiate that. You can also see that $f'(x) = sum_k=0^infty (k+1) x^k$, so what you want if $f'(1/10)$.
$endgroup$
– Clement C.
8 hours ago
$begingroup$
Well, if you forget the "high-school level" part (or are willing to be a bit hazy on the justification of the steps) you can always consider the function $f$ defined for $xin(-1,1)$ by $f(x) = sum_k=0^infty x^k+1$. You can compute a closed-form for it and differentiate that. You can also see that $f'(x) = sum_k=0^infty (k+1) x^k$, so what you want if $f'(1/10)$.
$endgroup$
– Clement C.
8 hours ago
1
1
$begingroup$
Possible duplicate of How can I evaluate $sum_n=0^infty(n+1)x^n$?
$endgroup$
– Martin R
7 hours ago
$begingroup$
Possible duplicate of How can I evaluate $sum_n=0^infty(n+1)x^n$?
$endgroup$
– Martin R
7 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Write out the infinite sum:
$$S=frac110^0+frac210^1+frac310^2+frac410^3+dots$$
Divide by ten and subtract from $S$:
$$frac110S=frac110^1+frac210^2+frac310^3+frac410^4+dots$$
$$S-frac110S=frac110^0+frac110^1+frac110^2+frac110^3+dots$$
This is a geometric series, whose sum can be easily calculated:
$$frac910S=frac11-1/10=frac109$$
$$S=frac10081$$
answered 8 hours ago
Parcly TaxelParcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
$endgroup$
add a comment |
$begingroup$
When $x|<1$ $$frac11-x=sum_k=0^infty x^k ~~~~(1).$$ Differentiate (1) w.r.t. $x$ to get $$frac1(1-x)^2= sum_k=0^infty k x^k-1 Rightarrow fracx.(1-x)^2= sum_k=0^infty k x^k~~~~(2)$$ Let us put $x=1/10$ in (1) and (2) and then add them to get $$sum_k=0^infty frack+110^k=frac109+frac1081=frac10081.$$
answered 7 hours ago
Dr Zafar Ahmed DScDr Zafar Ahmed DSc
2,7993 silver badges13 bronze badges
$endgroup$
add a comment |
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2 Answers
2
active
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2 Answers
2
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votes
$begingroup$
Write out the infinite sum:
$$S=frac110^0+frac210^1+frac310^2+frac410^3+dots$$
Divide by ten and subtract from $S$:
$$frac110S=frac110^1+frac210^2+frac310^3+frac410^4+dots$$
$$S-frac110S=frac110^0+frac110^1+frac110^2+frac110^3+dots$$
This is a geometric series, whose sum can be easily calculated:
$$frac910S=frac11-1/10=frac109$$
$$S=frac10081$$
answered 8 hours ago
Parcly TaxelParcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
$endgroup$
add a comment |
$begingroup$
Write out the infinite sum:
$$S=frac110^0+frac210^1+frac310^2+frac410^3+dots$$
Divide by ten and subtract from $S$:
$$frac110S=frac110^1+frac210^2+frac310^3+frac410^4+dots$$
$$S-frac110S=frac110^0+frac110^1+frac110^2+frac110^3+dots$$
This is a geometric series, whose sum can be easily calculated:
$$frac910S=frac11-1/10=frac109$$
$$S=frac10081$$
answered 8 hours ago
Parcly TaxelParcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
$endgroup$
add a comment |
$begingroup$
Write out the infinite sum:
$$S=frac110^0+frac210^1+frac310^2+frac410^3+dots$$
Divide by ten and subtract from $S$:
$$frac110S=frac110^1+frac210^2+frac310^3+frac410^4+dots$$
$$S-frac110S=frac110^0+frac110^1+frac110^2+frac110^3+dots$$
This is a geometric series, whose sum can be easily calculated:
$$frac910S=frac11-1/10=frac109$$
$$S=frac10081$$
answered 8 hours ago
Parcly TaxelParcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
$endgroup$
Write out the infinite sum:
$$S=frac110^0+frac210^1+frac310^2+frac410^3+dots$$
Divide by ten and subtract from $S$:
$$frac110S=frac110^1+frac210^2+frac310^3+frac410^4+dots$$
$$S-frac110S=frac110^0+frac110^1+frac110^2+frac110^3+dots$$
This is a geometric series, whose sum can be easily calculated:
$$frac910S=frac11-1/10=frac109$$
$$S=frac10081$$
answered 8 hours ago
Parcly TaxelParcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
answered 8 hours ago
Parcly TaxelParcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
answered 8 hours ago
Parcly TaxelParcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
answered 8 hours ago
Parcly TaxelParcly Taxel
48.2k13 gold badges77 silver badges117 bronze badges
48.2k13 gold badges77 silver badges117 bronze badges
add a comment |
add a comment |
$begingroup$
When $x|<1$ $$frac11-x=sum_k=0^infty x^k ~~~~(1).$$ Differentiate (1) w.r.t. $x$ to get $$frac1(1-x)^2= sum_k=0^infty k x^k-1 Rightarrow fracx.(1-x)^2= sum_k=0^infty k x^k~~~~(2)$$ Let us put $x=1/10$ in (1) and (2) and then add them to get $$sum_k=0^infty frack+110^k=frac109+frac1081=frac10081.$$
answered 7 hours ago
Dr Zafar Ahmed DScDr Zafar Ahmed DSc
2,7993 silver badges13 bronze badges
$endgroup$
add a comment |
$begingroup$
When $x|<1$ $$frac11-x=sum_k=0^infty x^k ~~~~(1).$$ Differentiate (1) w.r.t. $x$ to get $$frac1(1-x)^2= sum_k=0^infty k x^k-1 Rightarrow fracx.(1-x)^2= sum_k=0^infty k x^k~~~~(2)$$ Let us put $x=1/10$ in (1) and (2) and then add them to get $$sum_k=0^infty frack+110^k=frac109+frac1081=frac10081.$$
answered 7 hours ago
Dr Zafar Ahmed DScDr Zafar Ahmed DSc
2,7993 silver badges13 bronze badges
$endgroup$
add a comment |
$begingroup$
When $x|<1$ $$frac11-x=sum_k=0^infty x^k ~~~~(1).$$ Differentiate (1) w.r.t. $x$ to get $$frac1(1-x)^2= sum_k=0^infty k x^k-1 Rightarrow fracx.(1-x)^2= sum_k=0^infty k x^k~~~~(2)$$ Let us put $x=1/10$ in (1) and (2) and then add them to get $$sum_k=0^infty frack+110^k=frac109+frac1081=frac10081.$$
answered 7 hours ago
Dr Zafar Ahmed DScDr Zafar Ahmed DSc
2,7993 silver badges13 bronze badges
$endgroup$
When $x|<1$ $$frac11-x=sum_k=0^infty x^k ~~~~(1).$$ Differentiate (1) w.r.t. $x$ to get $$frac1(1-x)^2= sum_k=0^infty k x^k-1 Rightarrow fracx.(1-x)^2= sum_k=0^infty k x^k~~~~(2)$$ Let us put $x=1/10$ in (1) and (2) and then add them to get $$sum_k=0^infty frack+110^k=frac109+frac1081=frac10081.$$
answered 7 hours ago
Dr Zafar Ahmed DScDr Zafar Ahmed DSc
2,7993 silver badges13 bronze badges
answered 7 hours ago
Dr Zafar Ahmed DScDr Zafar Ahmed DSc
2,7993 silver badges13 bronze badges
answered 7 hours ago
Dr Zafar Ahmed DScDr Zafar Ahmed DSc
2,7993 silver badges13 bronze badges
answered 7 hours ago
Dr Zafar Ahmed DScDr Zafar Ahmed DSc
2,7993 silver badges13 bronze badges
2,7993 silver badges13 bronze badges
add a comment |
add a comment |
Jon9 is a new contributor. Be nice, and check out our Code of Conduct.
Jon9 is a new contributor. Be nice, and check out our Code of Conduct.
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1
$begingroup$
I'd by splitting it into the sum of $k/10^k$ and $1/10^k$, where the latter is easily computable as it is a geometric series.
$endgroup$
– Viktor Glombik
8 hours ago
3
$begingroup$
Well, if you forget the "high-school level" part (or are willing to be a bit hazy on the justification of the steps) you can always consider the function $f$ defined for $xin(-1,1)$ by $f(x) = sum_k=0^infty x^k+1$. You can compute a closed-form for it and differentiate that. You can also see that $f'(x) = sum_k=0^infty (k+1) x^k$, so what you want if $f'(1/10)$.
$endgroup$
– Clement C.
8 hours ago
1
$begingroup$
Possible duplicate of How can I evaluate $sum_n=0^infty(n+1)x^n$?
$endgroup$
– Martin R
7 hours ago