Evaluating $sumlimits_n ge 0 frac1x^2^n-y^2^n$ where $x, y in mathbb R^+$ and $x ne y, x>1.$
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I reduced a competition problem involving Fibbonacci numbers to the evaluation of this simple sum. I've tried telescoping, factorization, and even rewriting the product as
$$S = frac1x-y sum_n ge 0 prod_m=0^n-1 frac1x^2^m+y^2^m,$$
but none of these approaches led anywhere. I'm wondering if there is an elementary method for finding $S.$ I can't even solve the special case $y=1.$
sequences-and-series fibonacci-numbers infinite-product
asked Aug 29 at 23:27
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669211
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up vote
3
down vote
favorite
I reduced a competition problem involving Fibbonacci numbers to the evaluation of this simple sum. I've tried telescoping, factorization, and even rewriting the product as
$$S = frac1x-y sum_n ge 0 prod_m=0^n-1 frac1x^2^m+y^2^m,$$
but none of these approaches led anywhere. I'm wondering if there is an elementary method for finding $S.$ I can't even solve the special case $y=1.$
sequences-and-series fibonacci-numbers infinite-product
asked Aug 29 at 23:27
Display name
669211
What are $x,y$?
– Sisyphus
Aug 29 at 23:31
@Sisyphus I have edited the title to clarify.
– Display name
Aug 29 at 23:35
Not only $,y=1,$ is a very hard, actually $,y=0,$ is a challenge! $$sum_nge0x^2^n=,,? $$ Please adjust the convergence criteria, $xgt y$ is not sufficient.
– Hazem Orabi
Aug 30 at 10:15
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I reduced a competition problem involving Fibbonacci numbers to the evaluation of this simple sum. I've tried telescoping, factorization, and even rewriting the product as
$$S = frac1x-y sum_n ge 0 prod_m=0^n-1 frac1x^2^m+y^2^m,$$
but none of these approaches led anywhere. I'm wondering if there is an elementary method for finding $S.$ I can't even solve the special case $y=1.$
sequences-and-series fibonacci-numbers infinite-product
asked Aug 29 at 23:27
Display name
669211
I reduced a competition problem involving Fibbonacci numbers to the evaluation of this simple sum. I've tried telescoping, factorization, and even rewriting the product as
$$S = frac1x-y sum_n ge 0 prod_m=0^n-1 frac1x^2^m+y^2^m,$$
but none of these approaches led anywhere. I'm wondering if there is an elementary method for finding $S.$ I can't even solve the special case $y=1.$
sequences-and-series fibonacci-numbers infinite-product
sequences-and-series fibonacci-numbers infinite-product
asked Aug 29 at 23:27
Display name
669211
asked Aug 29 at 23:27
Display name
669211
edited Aug 30 at 19:47
asked Aug 29 at 23:27
Display name
669211
asked Aug 29 at 23:27
Display name
669211
asked Aug 29 at 23:27
Display name
669211
669211
What are $x,y$?
– Sisyphus
Aug 29 at 23:31
@Sisyphus I have edited the title to clarify.
– Display name
Aug 29 at 23:35
Not only $,y=1,$ is a very hard, actually $,y=0,$ is a challenge! $$sum_nge0x^2^n=,,? $$ Please adjust the convergence criteria, $xgt y$ is not sufficient.
– Hazem Orabi
Aug 30 at 10:15
add a comment |Â
What are $x,y$?
– Sisyphus
Aug 29 at 23:31
@Sisyphus I have edited the title to clarify.
– Display name
Aug 29 at 23:35
Not only $,y=1,$ is a very hard, actually $,y=0,$ is a challenge! $$sum_nge0x^2^n=,,? $$ Please adjust the convergence criteria, $xgt y$ is not sufficient.
– Hazem Orabi
Aug 30 at 10:15
What are $x,y$?
– Sisyphus
Aug 29 at 23:31
What are $x,y$?
– Sisyphus
Aug 29 at 23:31
@Sisyphus I have edited the title to clarify.
– Display name
Aug 29 at 23:35
@Sisyphus I have edited the title to clarify.
– Display name
Aug 29 at 23:35
Not only $,y=1,$ is a very hard, actually $,y=0,$ is a challenge! $$sum_nge0x^2^n=,,? $$ Please adjust the convergence criteria, $xgt y$ is not sufficient.
– Hazem Orabi
Aug 30 at 10:15
Not only $,y=1,$ is a very hard, actually $,y=0,$ is a challenge! $$sum_nge0x^2^n=,,? $$ Please adjust the convergence criteria, $xgt y$ is not sufficient.
– Hazem Orabi
Aug 30 at 10:15
add a comment |Â
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What are $x,y$?
– Sisyphus
Aug 29 at 23:31
@Sisyphus I have edited the title to clarify.
– Display name
Aug 29 at 23:35
Not only $,y=1,$ is a very hard, actually $,y=0,$ is a challenge! $$sum_nge0x^2^n=,,? $$ Please adjust the convergence criteria, $xgt y$ is not sufficient.
– Hazem Orabi
Aug 30 at 10:15