Sum of a finite arithmetic sequence [closed]

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Let $S_n = sum_j=0^n-12j(2j-1)x^2j$ where $|x| < 1$. What is the finite sum corresponds to?



Edit:
beginequation*
S_n = sum_j=0^n-12j(2j-1)x^2j\
= x^2 fracd^2dx^2left[frac1 - x^2n1-x^2right]
endequation*






calculus sequences-and-series arithmetic







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closed as off-topic by Henrik, amWhy, Paul Frost, user91500, Nosrati Sep 1 at 17:10


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." – Henrik, amWhy, Nosrati
If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    What are your thoughts on the problem? What have you tried?
    – Matt
    Sep 1 at 9:13










  • I can't think of a way of proceeding. Any hints would be great.
    – Shana
    Sep 1 at 9:14














up vote
-1
down vote

favorite
1












Let $S_n = sum_j=0^n-12j(2j-1)x^2j$ where $|x| < 1$. What is the finite sum corresponds to?



Edit:
beginequation*
S_n = sum_j=0^n-12j(2j-1)x^2j\
= x^2 fracd^2dx^2left[frac1 - x^2n1-x^2right]
endequation*






calculus sequences-and-series arithmetic







share|cite|improve this question















closed as off-topic by Henrik, amWhy, Paul Frost, user91500, Nosrati Sep 1 at 17:10


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." – Henrik, amWhy, Nosrati
If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    What are your thoughts on the problem? What have you tried?
    – Matt
    Sep 1 at 9:13










  • I can't think of a way of proceeding. Any hints would be great.
    – Shana
    Sep 1 at 9:14












up vote
-1
down vote

favorite
1









up vote
-1
down vote

favorite
1






1





Let $S_n = sum_j=0^n-12j(2j-1)x^2j$ where $|x| < 1$. What is the finite sum corresponds to?



Edit:
beginequation*
S_n = sum_j=0^n-12j(2j-1)x^2j\
= x^2 fracd^2dx^2left[frac1 - x^2n1-x^2right]
endequation*






calculus sequences-and-series arithmetic







share|cite|improve this question















Let $S_n = sum_j=0^n-12j(2j-1)x^2j$ where $|x| < 1$. What is the finite sum corresponds to?



Edit:
beginequation*
S_n = sum_j=0^n-12j(2j-1)x^2j\
= x^2 fracd^2dx^2left[frac1 - x^2n1-x^2right]
endequation*






calculus sequences-and-series arithmetic




calculus sequences-and-series arithmetic






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Sep 1 at 9:51

























asked Sep 1 at 9:12









Shana

408




408




closed as off-topic by Henrik, amWhy, Paul Frost, user91500, Nosrati Sep 1 at 17:10


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." – Henrik, amWhy, Nosrati
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Henrik, amWhy, Paul Frost, user91500, Nosrati Sep 1 at 17:10


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." – Henrik, amWhy, Nosrati
If this question can be reworded to fit the rules in the help center, please edit the question.







  • 1




    What are your thoughts on the problem? What have you tried?
    – Matt
    Sep 1 at 9:13










  • I can't think of a way of proceeding. Any hints would be great.
    – Shana
    Sep 1 at 9:14












  • 1




    What are your thoughts on the problem? What have you tried?
    – Matt
    Sep 1 at 9:13










  • I can't think of a way of proceeding. Any hints would be great.
    – Shana
    Sep 1 at 9:14







1




1




What are your thoughts on the problem? What have you tried?
– Matt
Sep 1 at 9:13




What are your thoughts on the problem? What have you tried?
– Matt
Sep 1 at 9:13












I can't think of a way of proceeding. Any hints would be great.
– Shana
Sep 1 at 9:14




I can't think of a way of proceeding. Any hints would be great.
– Shana
Sep 1 at 9:14










1 Answer
1






active

oldest

votes

















up vote
2
down vote



accepted










Note that
$$sum_j=0^n-12j(2j-1)x^2j=x^2fracd^2dx^2left(sum_j=0^n-1(x^2)^jright).$$
where the finite sum on the right has a closed formula that you should know (see LINK).



Can you take it from here? Please show your effort: edit your question and, below it, write about your progress toward the final answer!






share|cite|improve this answer






















  • Thanks for the hint. But still I can't think how to proceed this. Why you have written it like this?
    – Shana
    Sep 1 at 9:18











  • The sum on the right has a closed formula that you should know: en.wikipedia.org/wiki/Geometric_progression#Derivation
    – Robert Z
    Sep 1 at 9:30











  • I can solve the sum in RHS. But how did you come up with this overall equation?
    – Shana
    Sep 1 at 9:34










  • Brilliant idea! Thanks for sharing.
    – NoChance
    Sep 1 at 9:35










  • @RobertZ: I think I got the logic. Thanks.
    – Shana
    Sep 1 at 9:37

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
2
down vote



accepted










Note that
$$sum_j=0^n-12j(2j-1)x^2j=x^2fracd^2dx^2left(sum_j=0^n-1(x^2)^jright).$$
where the finite sum on the right has a closed formula that you should know (see LINK).



Can you take it from here? Please show your effort: edit your question and, below it, write about your progress toward the final answer!






share|cite|improve this answer






















  • Thanks for the hint. But still I can't think how to proceed this. Why you have written it like this?
    – Shana
    Sep 1 at 9:18











  • The sum on the right has a closed formula that you should know: en.wikipedia.org/wiki/Geometric_progression#Derivation
    – Robert Z
    Sep 1 at 9:30











  • I can solve the sum in RHS. But how did you come up with this overall equation?
    – Shana
    Sep 1 at 9:34










  • Brilliant idea! Thanks for sharing.
    – NoChance
    Sep 1 at 9:35










  • @RobertZ: I think I got the logic. Thanks.
    – Shana
    Sep 1 at 9:37














up vote
2
down vote



accepted










Note that
$$sum_j=0^n-12j(2j-1)x^2j=x^2fracd^2dx^2left(sum_j=0^n-1(x^2)^jright).$$
where the finite sum on the right has a closed formula that you should know (see LINK).



Can you take it from here? Please show your effort: edit your question and, below it, write about your progress toward the final answer!






share|cite|improve this answer






















  • Thanks for the hint. But still I can't think how to proceed this. Why you have written it like this?
    – Shana
    Sep 1 at 9:18











  • The sum on the right has a closed formula that you should know: en.wikipedia.org/wiki/Geometric_progression#Derivation
    – Robert Z
    Sep 1 at 9:30











  • I can solve the sum in RHS. But how did you come up with this overall equation?
    – Shana
    Sep 1 at 9:34










  • Brilliant idea! Thanks for sharing.
    – NoChance
    Sep 1 at 9:35










  • @RobertZ: I think I got the logic. Thanks.
    – Shana
    Sep 1 at 9:37












up vote
2
down vote



accepted







up vote
2
down vote



accepted






Note that
$$sum_j=0^n-12j(2j-1)x^2j=x^2fracd^2dx^2left(sum_j=0^n-1(x^2)^jright).$$
where the finite sum on the right has a closed formula that you should know (see LINK).



Can you take it from here? Please show your effort: edit your question and, below it, write about your progress toward the final answer!






share|cite|improve this answer














Note that
$$sum_j=0^n-12j(2j-1)x^2j=x^2fracd^2dx^2left(sum_j=0^n-1(x^2)^jright).$$
where the finite sum on the right has a closed formula that you should know (see LINK).



Can you take it from here? Please show your effort: edit your question and, below it, write about your progress toward the final answer!







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Sep 1 at 9:54

























answered Sep 1 at 9:14









Robert Z

85.6k1055123




85.6k1055123











  • Thanks for the hint. But still I can't think how to proceed this. Why you have written it like this?
    – Shana
    Sep 1 at 9:18











  • The sum on the right has a closed formula that you should know: en.wikipedia.org/wiki/Geometric_progression#Derivation
    – Robert Z
    Sep 1 at 9:30











  • I can solve the sum in RHS. But how did you come up with this overall equation?
    – Shana
    Sep 1 at 9:34










  • Brilliant idea! Thanks for sharing.
    – NoChance
    Sep 1 at 9:35










  • @RobertZ: I think I got the logic. Thanks.
    – Shana
    Sep 1 at 9:37
















  • Thanks for the hint. But still I can't think how to proceed this. Why you have written it like this?
    – Shana
    Sep 1 at 9:18











  • The sum on the right has a closed formula that you should know: en.wikipedia.org/wiki/Geometric_progression#Derivation
    – Robert Z
    Sep 1 at 9:30











  • I can solve the sum in RHS. But how did you come up with this overall equation?
    – Shana
    Sep 1 at 9:34










  • Brilliant idea! Thanks for sharing.
    – NoChance
    Sep 1 at 9:35










  • @RobertZ: I think I got the logic. Thanks.
    – Shana
    Sep 1 at 9:37















Thanks for the hint. But still I can't think how to proceed this. Why you have written it like this?
– Shana
Sep 1 at 9:18





Thanks for the hint. But still I can't think how to proceed this. Why you have written it like this?
– Shana
Sep 1 at 9:18













The sum on the right has a closed formula that you should know: en.wikipedia.org/wiki/Geometric_progression#Derivation
– Robert Z
Sep 1 at 9:30





The sum on the right has a closed formula that you should know: en.wikipedia.org/wiki/Geometric_progression#Derivation
– Robert Z
Sep 1 at 9:30













I can solve the sum in RHS. But how did you come up with this overall equation?
– Shana
Sep 1 at 9:34




I can solve the sum in RHS. But how did you come up with this overall equation?
– Shana
Sep 1 at 9:34












Brilliant idea! Thanks for sharing.
– NoChance
Sep 1 at 9:35




Brilliant idea! Thanks for sharing.
– NoChance
Sep 1 at 9:35












@RobertZ: I think I got the logic. Thanks.
– Shana
Sep 1 at 9:37




@RobertZ: I think I got the logic. Thanks.
– Shana
Sep 1 at 9:37


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