How to find sum of the infinite series $frac{n^2}{n!}$ [closed]
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I have tried the Riemann sum also but couldn't find the answer.
real-analysis sequences-and-series
edited Nov 18 at 9:47
Joey Kilpatrick
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asked Nov 18 at 9:18
Ravi Satpute
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closed as off-topic by amWhy, Shailesh, max_zorn, Hanul Jeon, Brahadeesh Nov 18 at 11:50
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up vote
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I have tried the Riemann sum also but couldn't find the answer.
real-analysis sequences-and-series
edited Nov 18 at 9:47
Joey Kilpatrick
1,183422
asked Nov 18 at 9:18
Ravi Satpute
84
closed as off-topic by amWhy, Shailesh, max_zorn, Hanul Jeon, Brahadeesh Nov 18 at 11:50
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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Shailesh, max_zorn, Hanul Jeon, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:21
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up vote
1
down vote
favorite
I have tried the Riemann sum also but couldn't find the answer.
real-analysis sequences-and-series
edited Nov 18 at 9:47
Joey Kilpatrick
1,183422
asked Nov 18 at 9:18
Ravi Satpute
84
I have tried the Riemann sum also but couldn't find the answer.
real-analysis sequences-and-series
real-analysis sequences-and-series
edited Nov 18 at 9:47
Joey Kilpatrick
1,183422
asked Nov 18 at 9:18
Ravi Satpute
84
edited Nov 18 at 9:47
Joey Kilpatrick
1,183422
asked Nov 18 at 9:18
Ravi Satpute
84
edited Nov 18 at 9:47
Joey Kilpatrick
1,183422
edited Nov 18 at 9:47
Joey Kilpatrick
1,183422
edited Nov 18 at 9:47
Joey Kilpatrick
1,183422
1,183422
asked Nov 18 at 9:18
Ravi Satpute
84
asked Nov 18 at 9:18
Ravi Satpute
84
asked Nov 18 at 9:18
Ravi Satpute
84
84
closed as off-topic by amWhy, Shailesh, max_zorn, Hanul Jeon, Brahadeesh Nov 18 at 11:50
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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Shailesh, max_zorn, Hanul Jeon, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by amWhy, Shailesh, max_zorn, Hanul Jeon, Brahadeesh Nov 18 at 11:50
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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Shailesh, max_zorn, Hanul Jeon, Brahadeesh
If this question can be reworded to fit the rules in the help center, please edit the question.
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:21
add a comment |
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:21
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:21
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:21
add a comment |
2 Answers
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1
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accepted
Note that
- $sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!}$
- $e^x =sum_{n=0}^{infty}frac{x^n}{n!} Rightarrow xe^x = sum_{n=0}^{infty}frac{x^{n+1}}{n!}$
- $left( xe^x right)' = (1+x)e^x = sum^{infty}_{color{blue}{n=0}}frac{(n+1)x^{n}}{n!} stackrel{i = n+1}{=} sum^{infty}_{color{blue}{i=1}}frac{ix^{i-1}}{(i-1)!}$
$$Rightarrow sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!} = (1+1)e = 2e$$
answered Nov 18 at 10:48
trancelocation
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Hint
Consider
$$S=sum_{n=0}^infty frac {n^2}{n!}x^n=sum_{n=0}^infty frac {n(n-1)+n}{n!}x^n=sum_{n=0}^infty frac {n(n-1)}{n!}x^n+sum_{n=0}^infty frac {n}{n!}x^n$$ that is to say
$$S=x^2sum_{n=0}^infty frac {n(n-1)}{n!}x^{n-2}+xsum_{n=0}^infty frac {n}{n!}x^{n-1}$$
When finished, let $x=1$.
answered Nov 18 at 9:32
Claude Leibovici
117k1156131
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Note that
- $sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!}$
- $e^x =sum_{n=0}^{infty}frac{x^n}{n!} Rightarrow xe^x = sum_{n=0}^{infty}frac{x^{n+1}}{n!}$
- $left( xe^x right)' = (1+x)e^x = sum^{infty}_{color{blue}{n=0}}frac{(n+1)x^{n}}{n!} stackrel{i = n+1}{=} sum^{infty}_{color{blue}{i=1}}frac{ix^{i-1}}{(i-1)!}$
$$Rightarrow sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!} = (1+1)e = 2e$$
answered Nov 18 at 10:48
trancelocation
8,6321520
add a comment |
up vote
1
down vote
accepted
Note that
- $sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!}$
- $e^x =sum_{n=0}^{infty}frac{x^n}{n!} Rightarrow xe^x = sum_{n=0}^{infty}frac{x^{n+1}}{n!}$
- $left( xe^x right)' = (1+x)e^x = sum^{infty}_{color{blue}{n=0}}frac{(n+1)x^{n}}{n!} stackrel{i = n+1}{=} sum^{infty}_{color{blue}{i=1}}frac{ix^{i-1}}{(i-1)!}$
$$Rightarrow sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!} = (1+1)e = 2e$$
answered Nov 18 at 10:48
trancelocation
8,6321520
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Note that
- $sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!}$
- $e^x =sum_{n=0}^{infty}frac{x^n}{n!} Rightarrow xe^x = sum_{n=0}^{infty}frac{x^{n+1}}{n!}$
- $left( xe^x right)' = (1+x)e^x = sum^{infty}_{color{blue}{n=0}}frac{(n+1)x^{n}}{n!} stackrel{i = n+1}{=} sum^{infty}_{color{blue}{i=1}}frac{ix^{i-1}}{(i-1)!}$
$$Rightarrow sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!} = (1+1)e = 2e$$
answered Nov 18 at 10:48
trancelocation
8,6321520
Note that
- $sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!}$
- $e^x =sum_{n=0}^{infty}frac{x^n}{n!} Rightarrow xe^x = sum_{n=0}^{infty}frac{x^{n+1}}{n!}$
- $left( xe^x right)' = (1+x)e^x = sum^{infty}_{color{blue}{n=0}}frac{(n+1)x^{n}}{n!} stackrel{i = n+1}{=} sum^{infty}_{color{blue}{i=1}}frac{ix^{i-1}}{(i-1)!}$
$$Rightarrow sum_{n=1}^{infty}frac{n^2}{n!} = sum_{n=1}^{infty}frac{n}{(n-1)!} = (1+1)e = 2e$$
answered Nov 18 at 10:48
trancelocation
8,6321520
answered Nov 18 at 10:48
trancelocation
8,6321520
answered Nov 18 at 10:48
trancelocation
8,6321520
answered Nov 18 at 10:48
trancelocation
8,6321520
8,6321520
add a comment |
add a comment |
up vote
6
down vote
Hint
Consider
$$S=sum_{n=0}^infty frac {n^2}{n!}x^n=sum_{n=0}^infty frac {n(n-1)+n}{n!}x^n=sum_{n=0}^infty frac {n(n-1)}{n!}x^n+sum_{n=0}^infty frac {n}{n!}x^n$$ that is to say
$$S=x^2sum_{n=0}^infty frac {n(n-1)}{n!}x^{n-2}+xsum_{n=0}^infty frac {n}{n!}x^{n-1}$$
When finished, let $x=1$.
answered Nov 18 at 9:32
Claude Leibovici
117k1156131
add a comment |
up vote
6
down vote
Hint
Consider
$$S=sum_{n=0}^infty frac {n^2}{n!}x^n=sum_{n=0}^infty frac {n(n-1)+n}{n!}x^n=sum_{n=0}^infty frac {n(n-1)}{n!}x^n+sum_{n=0}^infty frac {n}{n!}x^n$$ that is to say
$$S=x^2sum_{n=0}^infty frac {n(n-1)}{n!}x^{n-2}+xsum_{n=0}^infty frac {n}{n!}x^{n-1}$$
When finished, let $x=1$.
answered Nov 18 at 9:32
Claude Leibovici
117k1156131
add a comment |
up vote
6
down vote
up vote
6
down vote
Hint
Consider
$$S=sum_{n=0}^infty frac {n^2}{n!}x^n=sum_{n=0}^infty frac {n(n-1)+n}{n!}x^n=sum_{n=0}^infty frac {n(n-1)}{n!}x^n+sum_{n=0}^infty frac {n}{n!}x^n$$ that is to say
$$S=x^2sum_{n=0}^infty frac {n(n-1)}{n!}x^{n-2}+xsum_{n=0}^infty frac {n}{n!}x^{n-1}$$
When finished, let $x=1$.
answered Nov 18 at 9:32
Claude Leibovici
117k1156131
Hint
Consider
$$S=sum_{n=0}^infty frac {n^2}{n!}x^n=sum_{n=0}^infty frac {n(n-1)+n}{n!}x^n=sum_{n=0}^infty frac {n(n-1)}{n!}x^n+sum_{n=0}^infty frac {n}{n!}x^n$$ that is to say
$$S=x^2sum_{n=0}^infty frac {n(n-1)}{n!}x^{n-2}+xsum_{n=0}^infty frac {n}{n!}x^{n-1}$$
When finished, let $x=1$.
answered Nov 18 at 9:32
Claude Leibovici
117k1156131
answered Nov 18 at 9:32
Claude Leibovici
117k1156131
answered Nov 18 at 9:32
Claude Leibovici
117k1156131
answered Nov 18 at 9:32
Claude Leibovici
117k1156131
117k1156131
add a comment |
add a comment |
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Nov 18 at 9:21