Sum of the first 50 elements of the sum $sum n a^n$
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0
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Find the sum of the first 50 elements in for $x = 1.5$
$$sum_n n x^n$$
Now, if I were to find an infinite convergant sum, I would know what to do, since
$$frac{1}{1-x} = sum_nx^n$$
it's enough to differentiate this and perform some numerical manipulations.
However, as I am being asked to find the sum of a certain number of terms, I don't know how to proceed.
sequences-and-series discrete-mathematics
asked Nov 14 at 10:57
Aemilius
1,664314
add a comment |
up vote
0
down vote
favorite
Find the sum of the first 50 elements in for $x = 1.5$
$$sum_n n x^n$$
Now, if I were to find an infinite convergant sum, I would know what to do, since
$$frac{1}{1-x} = sum_nx^n$$
it's enough to differentiate this and perform some numerical manipulations.
However, as I am being asked to find the sum of a certain number of terms, I don't know how to proceed.
sequences-and-series discrete-mathematics
asked Nov 14 at 10:57
Aemilius
1,664314
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Find the sum of the first 50 elements in for $x = 1.5$
$$sum_n n x^n$$
Now, if I were to find an infinite convergant sum, I would know what to do, since
$$frac{1}{1-x} = sum_nx^n$$
it's enough to differentiate this and perform some numerical manipulations.
However, as I am being asked to find the sum of a certain number of terms, I don't know how to proceed.
sequences-and-series discrete-mathematics
asked Nov 14 at 10:57
Aemilius
1,664314
Find the sum of the first 50 elements in for $x = 1.5$
$$sum_n n x^n$$
Now, if I were to find an infinite convergant sum, I would know what to do, since
$$frac{1}{1-x} = sum_nx^n$$
it's enough to differentiate this and perform some numerical manipulations.
However, as I am being asked to find the sum of a certain number of terms, I don't know how to proceed.
sequences-and-series discrete-mathematics
sequences-and-series discrete-mathematics
asked Nov 14 at 10:57
Aemilius
1,664314
asked Nov 14 at 10:57
Aemilius
1,664314
asked Nov 14 at 10:57
Aemilius
1,664314
asked Nov 14 at 10:57
Aemilius
1,664314
asked Nov 14 at 10:57
Aemilius
1,664314
1,664314
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
Hint: evaluate
$$
xpartial_xsum_n x^n
$$
Also, as is well known,
$$
sum_{n = 0}^N x^n = frac{1-x^{N+1}}{1-x}
$$
answered Nov 14 at 11:02
denklo
3955
The second part of your answer actually solved my problem, thank you! What does this symbol $partial_x$ mean in the first part?
– Aemilius
Nov 14 at 11:07
@Aemilius, its the derivative wrt. x. Are you sure that you only need the second part? In your question the sum to evaluate is not the geometirc series, but $$sum_{n=0}^{50} mathbf{n }x^n$$
– denklo
Nov 14 at 11:08
@Aemilius or is this a typo?
– denklo
Nov 14 at 11:10
How did you come up with the idea to use the sum containing the partial derivative? Intuition?
– Aemilius
Nov 14 at 11:10
Yes, but I can differentiate what you have written and obtain my result, or at least I guess so.
– Aemilius
Nov 14 at 11:11
|
show 2 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint: evaluate
$$
xpartial_xsum_n x^n
$$
Also, as is well known,
$$
sum_{n = 0}^N x^n = frac{1-x^{N+1}}{1-x}
$$
answered Nov 14 at 11:02
denklo
3955
The second part of your answer actually solved my problem, thank you! What does this symbol $partial_x$ mean in the first part?
– Aemilius
Nov 14 at 11:07
@Aemilius, its the derivative wrt. x. Are you sure that you only need the second part? In your question the sum to evaluate is not the geometirc series, but $$sum_{n=0}^{50} mathbf{n }x^n$$
– denklo
Nov 14 at 11:08
@Aemilius or is this a typo?
– denklo
Nov 14 at 11:10
How did you come up with the idea to use the sum containing the partial derivative? Intuition?
– Aemilius
Nov 14 at 11:10
Yes, but I can differentiate what you have written and obtain my result, or at least I guess so.
– Aemilius
Nov 14 at 11:11
|
show 2 more comments
up vote
1
down vote
Hint: evaluate
$$
xpartial_xsum_n x^n
$$
Also, as is well known,
$$
sum_{n = 0}^N x^n = frac{1-x^{N+1}}{1-x}
$$
answered Nov 14 at 11:02
denklo
3955
The second part of your answer actually solved my problem, thank you! What does this symbol $partial_x$ mean in the first part?
– Aemilius
Nov 14 at 11:07
@Aemilius, its the derivative wrt. x. Are you sure that you only need the second part? In your question the sum to evaluate is not the geometirc series, but $$sum_{n=0}^{50} mathbf{n }x^n$$
– denklo
Nov 14 at 11:08
@Aemilius or is this a typo?
– denklo
Nov 14 at 11:10
How did you come up with the idea to use the sum containing the partial derivative? Intuition?
– Aemilius
Nov 14 at 11:10
Yes, but I can differentiate what you have written and obtain my result, or at least I guess so.
– Aemilius
Nov 14 at 11:11
|
show 2 more comments
up vote
1
down vote
up vote
1
down vote
Hint: evaluate
$$
xpartial_xsum_n x^n
$$
Also, as is well known,
$$
sum_{n = 0}^N x^n = frac{1-x^{N+1}}{1-x}
$$
answered Nov 14 at 11:02
denklo
3955
Hint: evaluate
$$
xpartial_xsum_n x^n
$$
Also, as is well known,
$$
sum_{n = 0}^N x^n = frac{1-x^{N+1}}{1-x}
$$
answered Nov 14 at 11:02
denklo
3955
answered Nov 14 at 11:02
denklo
3955
answered Nov 14 at 11:02
denklo
3955
answered Nov 14 at 11:02
denklo
3955
3955
The second part of your answer actually solved my problem, thank you! What does this symbol $partial_x$ mean in the first part?
– Aemilius
Nov 14 at 11:07
@Aemilius, its the derivative wrt. x. Are you sure that you only need the second part? In your question the sum to evaluate is not the geometirc series, but $$sum_{n=0}^{50} mathbf{n }x^n$$
– denklo
Nov 14 at 11:08
@Aemilius or is this a typo?
– denklo
Nov 14 at 11:10
How did you come up with the idea to use the sum containing the partial derivative? Intuition?
– Aemilius
Nov 14 at 11:10
Yes, but I can differentiate what you have written and obtain my result, or at least I guess so.
– Aemilius
Nov 14 at 11:11
|
show 2 more comments
The second part of your answer actually solved my problem, thank you! What does this symbol $partial_x$ mean in the first part?
– Aemilius
Nov 14 at 11:07
@Aemilius, its the derivative wrt. x. Are you sure that you only need the second part? In your question the sum to evaluate is not the geometirc series, but $$sum_{n=0}^{50} mathbf{n }x^n$$
– denklo
Nov 14 at 11:08
@Aemilius or is this a typo?
– denklo
Nov 14 at 11:10
How did you come up with the idea to use the sum containing the partial derivative? Intuition?
– Aemilius
Nov 14 at 11:10
Yes, but I can differentiate what you have written and obtain my result, or at least I guess so.
– Aemilius
Nov 14 at 11:11
The second part of your answer actually solved my problem, thank you! What does this symbol $partial_x$ mean in the first part?
– Aemilius
Nov 14 at 11:07
The second part of your answer actually solved my problem, thank you! What does this symbol $partial_x$ mean in the first part?
– Aemilius
Nov 14 at 11:07
@Aemilius, its the derivative wrt. x. Are you sure that you only need the second part? In your question the sum to evaluate is not the geometirc series, but $$sum_{n=0}^{50} mathbf{n }x^n$$
– denklo
Nov 14 at 11:08
@Aemilius, its the derivative wrt. x. Are you sure that you only need the second part? In your question the sum to evaluate is not the geometirc series, but $$sum_{n=0}^{50} mathbf{n }x^n$$
– denklo
Nov 14 at 11:08
@Aemilius or is this a typo?
– denklo
Nov 14 at 11:10
@Aemilius or is this a typo?
– denklo
Nov 14 at 11:10
How did you come up with the idea to use the sum containing the partial derivative? Intuition?
– Aemilius
Nov 14 at 11:10
How did you come up with the idea to use the sum containing the partial derivative? Intuition?
– Aemilius
Nov 14 at 11:10
Yes, but I can differentiate what you have written and obtain my result, or at least I guess so.
– Aemilius
Nov 14 at 11:11
Yes, but I can differentiate what you have written and obtain my result, or at least I guess so.
– Aemilius
Nov 14 at 11:11
|
show 2 more comments
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