Evaluate $sum_{n=1}^{infty} frac{n^2}{3^n}$ [closed]
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Evaluate $sum_{n=1}^{infty} frac{n^2}{3^n}$. I know how to solve for $sum_{n=1}^{infty} frac{n}{3^n}$ but not sure what the trick should be for the $n^2$ case.
real-analysis sequences-and-series
asked Jan 8 at 3:12
Daniel LiDaniel Li
594312
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closed as off-topic by Eevee Trainer, Nosrati, RRL, Mark Viola, Hans Lundmark Jan 8 at 5:43
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Nosrati, RRL, Mark Viola
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
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Evaluate $sum_{n=1}^{infty} frac{n^2}{3^n}$. I know how to solve for $sum_{n=1}^{infty} frac{n}{3^n}$ but not sure what the trick should be for the $n^2$ case.
real-analysis sequences-and-series
asked Jan 8 at 3:12
Daniel LiDaniel Li
594312
$endgroup$
closed as off-topic by Eevee Trainer, Nosrati, RRL, Mark Viola, Hans Lundmark Jan 8 at 5:43
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Nosrati, RRL, Mark Viola
If this question can be reworded to fit the rules in the help center, please edit the question.
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Possible duplicate of this. The solution to the above problem is given in the "added note" in the answer by Eric Naslund of the linked post.
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– Mason
Jan 8 at 3:22
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math.stackexchange.com/questions/593996/…
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– lab bhattacharjee
Jan 8 at 4:07
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Possible duplicate of How to prove $sum_{n=0}^{infty} frac{n^2}{2^n} = 6$?
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– Hans Lundmark
Jan 8 at 5:43
add a comment |
$begingroup$
Evaluate $sum_{n=1}^{infty} frac{n^2}{3^n}$. I know how to solve for $sum_{n=1}^{infty} frac{n}{3^n}$ but not sure what the trick should be for the $n^2$ case.
real-analysis sequences-and-series
asked Jan 8 at 3:12
Daniel LiDaniel Li
594312
$endgroup$
Evaluate $sum_{n=1}^{infty} frac{n^2}{3^n}$. I know how to solve for $sum_{n=1}^{infty} frac{n}{3^n}$ but not sure what the trick should be for the $n^2$ case.
real-analysis sequences-and-series
real-analysis sequences-and-series
asked Jan 8 at 3:12
Daniel LiDaniel Li
594312
asked Jan 8 at 3:12
Daniel LiDaniel Li
594312
asked Jan 8 at 3:12
Daniel LiDaniel Li
594312
asked Jan 8 at 3:12
Daniel LiDaniel Li
594312
asked Jan 8 at 3:12
Daniel LiDaniel Li
594312
594312
closed as off-topic by Eevee Trainer, Nosrati, RRL, Mark Viola, Hans Lundmark Jan 8 at 5:43
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Nosrati, RRL, Mark Viola
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Eevee Trainer, Nosrati, RRL, Mark Viola, Hans Lundmark Jan 8 at 5:43
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Nosrati, RRL, Mark Viola
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Possible duplicate of this. The solution to the above problem is given in the "added note" in the answer by Eric Naslund of the linked post.
$endgroup$
– Mason
Jan 8 at 3:22
$begingroup$
math.stackexchange.com/questions/593996/…
$endgroup$
– lab bhattacharjee
Jan 8 at 4:07
$begingroup$
Possible duplicate of How to prove $sum_{n=0}^{infty} frac{n^2}{2^n} = 6$?
$endgroup$
– Hans Lundmark
Jan 8 at 5:43
add a comment |
$begingroup$
Possible duplicate of this. The solution to the above problem is given in the "added note" in the answer by Eric Naslund of the linked post.
$endgroup$
– Mason
Jan 8 at 3:22
$begingroup$
math.stackexchange.com/questions/593996/…
$endgroup$
– lab bhattacharjee
Jan 8 at 4:07
$begingroup$
Possible duplicate of How to prove $sum_{n=0}^{infty} frac{n^2}{2^n} = 6$?
$endgroup$
– Hans Lundmark
Jan 8 at 5:43
$begingroup$
Possible duplicate of this. The solution to the above problem is given in the "added note" in the answer by Eric Naslund of the linked post.
$endgroup$
– Mason
Jan 8 at 3:22
$begingroup$
Possible duplicate of this. The solution to the above problem is given in the "added note" in the answer by Eric Naslund of the linked post.
$endgroup$
– Mason
Jan 8 at 3:22
$begingroup$
math.stackexchange.com/questions/593996/…
$endgroup$
– lab bhattacharjee
Jan 8 at 4:07
$begingroup$
math.stackexchange.com/questions/593996/…
$endgroup$
– lab bhattacharjee
Jan 8 at 4:07
$begingroup$
Possible duplicate of How to prove $sum_{n=0}^{infty} frac{n^2}{2^n} = 6$?
$endgroup$
– Hans Lundmark
Jan 8 at 5:43
$begingroup$
Possible duplicate of How to prove $sum_{n=0}^{infty} frac{n^2}{2^n} = 6$?
$endgroup$
– Hans Lundmark
Jan 8 at 5:43
add a comment |
1 Answer
1
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oldest
votes
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The trick is to write
$$n^2=n(n-1)+n$$
$$n^3=n(n-1)(n-2)+3n(n-1)+n$$
$$n^4=n(n-1)(n-2)(n-3)+6n(n-1)(n-2)+7n(n-1)+n$$ and so on.
answered Jan 8 at 3:42
Claude LeiboviciClaude Leibovici
120k1157132
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The trick is to write
$$n^2=n(n-1)+n$$
$$n^3=n(n-1)(n-2)+3n(n-1)+n$$
$$n^4=n(n-1)(n-2)(n-3)+6n(n-1)(n-2)+7n(n-1)+n$$ and so on.
answered Jan 8 at 3:42
Claude LeiboviciClaude Leibovici
120k1157132
$endgroup$
add a comment |
$begingroup$
The trick is to write
$$n^2=n(n-1)+n$$
$$n^3=n(n-1)(n-2)+3n(n-1)+n$$
$$n^4=n(n-1)(n-2)(n-3)+6n(n-1)(n-2)+7n(n-1)+n$$ and so on.
answered Jan 8 at 3:42
Claude LeiboviciClaude Leibovici
120k1157132
$endgroup$
add a comment |
$begingroup$
The trick is to write
$$n^2=n(n-1)+n$$
$$n^3=n(n-1)(n-2)+3n(n-1)+n$$
$$n^4=n(n-1)(n-2)(n-3)+6n(n-1)(n-2)+7n(n-1)+n$$ and so on.
answered Jan 8 at 3:42
Claude LeiboviciClaude Leibovici
120k1157132
$endgroup$
The trick is to write
$$n^2=n(n-1)+n$$
$$n^3=n(n-1)(n-2)+3n(n-1)+n$$
$$n^4=n(n-1)(n-2)(n-3)+6n(n-1)(n-2)+7n(n-1)+n$$ and so on.
answered Jan 8 at 3:42
Claude LeiboviciClaude Leibovici
120k1157132
answered Jan 8 at 3:42
Claude LeiboviciClaude Leibovici
120k1157132
answered Jan 8 at 3:42
Claude LeiboviciClaude Leibovici
120k1157132
answered Jan 8 at 3:42
Claude LeiboviciClaude Leibovici
120k1157132
120k1157132
add a comment |
add a comment |
$begingroup$
Possible duplicate of this. The solution to the above problem is given in the "added note" in the answer by Eric Naslund of the linked post.
$endgroup$
– Mason
Jan 8 at 3:22
$begingroup$
math.stackexchange.com/questions/593996/…
$endgroup$
– lab bhattacharjee
Jan 8 at 4:07
$begingroup$
Possible duplicate of How to prove $sum_{n=0}^{infty} frac{n^2}{2^n} = 6$?
$endgroup$
– Hans Lundmark
Jan 8 at 5:43