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Calculate the sum of geometrical progression
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I have the following progression
$$
frac11+x^2 + frac1(1+x^2)^2 + ... + frac1(1+x^2)^n
$$
I have that $a=frac11+x^2$ and $q=frac11+x^2$, then using $afrac1-q^n+11-q$ I got $frac(1+x^2)^n+1 - 1x^2(1+x^2)^n+1$
My solution seems to be a bit hard and I think that it's possible to simplify the solution above. I will be grateful for any help you can provide.
geometric-progressions
asked Jul 19 at 14:46
E. Shcherbo
525
 |Â
show 5 more comments
up vote
0
down vote
favorite
I have the following progression
$$
frac11+x^2 + frac1(1+x^2)^2 + ... + frac1(1+x^2)^n
$$
I have that $a=frac11+x^2$ and $q=frac11+x^2$, then using $afrac1-q^n+11-q$ I got $frac(1+x^2)^n+1 - 1x^2(1+x^2)^n+1$
My solution seems to be a bit hard and I think that it's possible to simplify the solution above. I will be grateful for any help you can provide.
geometric-progressions
asked Jul 19 at 14:46
E. Shcherbo
525
1
I suppose you could reduce it to $$frac1x^2-frac1x^2(1+x^2)^n+1$$
– TheSimpliFire
Jul 19 at 14:49
@TheSimpliFire, yes, I did it. So do you think that there is no another way to write the sum using this formula?
– E. Shcherbo
Jul 19 at 14:51
No, I don't think so. The expression looks fine by itself. Why do you want such a simplification anyway?
– TheSimpliFire
Jul 19 at 14:53
@TheSimpliFire, When I haven't an answer a simple solution says to me that I solved it correctly :-)
– E. Shcherbo
Jul 19 at 14:55
This works for $xneq 0$
– Rumpelstiltskin
Jul 19 at 14:56
 |Â
show 5 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have the following progression
$$
frac11+x^2 + frac1(1+x^2)^2 + ... + frac1(1+x^2)^n
$$
I have that $a=frac11+x^2$ and $q=frac11+x^2$, then using $afrac1-q^n+11-q$ I got $frac(1+x^2)^n+1 - 1x^2(1+x^2)^n+1$
My solution seems to be a bit hard and I think that it's possible to simplify the solution above. I will be grateful for any help you can provide.
geometric-progressions
asked Jul 19 at 14:46
E. Shcherbo
525
I have the following progression
$$
frac11+x^2 + frac1(1+x^2)^2 + ... + frac1(1+x^2)^n
$$
I have that $a=frac11+x^2$ and $q=frac11+x^2$, then using $afrac1-q^n+11-q$ I got $frac(1+x^2)^n+1 - 1x^2(1+x^2)^n+1$
My solution seems to be a bit hard and I think that it's possible to simplify the solution above. I will be grateful for any help you can provide.
geometric-progressions
asked Jul 19 at 14:46
E. Shcherbo
525
asked Jul 19 at 14:46
E. Shcherbo
525
asked Jul 19 at 14:46
E. Shcherbo
525
asked Jul 19 at 14:46
E. Shcherbo
525
525
1
I suppose you could reduce it to $$frac1x^2-frac1x^2(1+x^2)^n+1$$
– TheSimpliFire
Jul 19 at 14:49
@TheSimpliFire, yes, I did it. So do you think that there is no another way to write the sum using this formula?
– E. Shcherbo
Jul 19 at 14:51
No, I don't think so. The expression looks fine by itself. Why do you want such a simplification anyway?
– TheSimpliFire
Jul 19 at 14:53
@TheSimpliFire, When I haven't an answer a simple solution says to me that I solved it correctly :-)
– E. Shcherbo
Jul 19 at 14:55
This works for $xneq 0$
– Rumpelstiltskin
Jul 19 at 14:56
 |Â
show 5 more comments
1
I suppose you could reduce it to $$frac1x^2-frac1x^2(1+x^2)^n+1$$
– TheSimpliFire
Jul 19 at 14:49
@TheSimpliFire, yes, I did it. So do you think that there is no another way to write the sum using this formula?
– E. Shcherbo
Jul 19 at 14:51
No, I don't think so. The expression looks fine by itself. Why do you want such a simplification anyway?
– TheSimpliFire
Jul 19 at 14:53
@TheSimpliFire, When I haven't an answer a simple solution says to me that I solved it correctly :-)
– E. Shcherbo
Jul 19 at 14:55
This works for $xneq 0$
– Rumpelstiltskin
Jul 19 at 14:56
1
1
I suppose you could reduce it to $$frac1x^2-frac1x^2(1+x^2)^n+1$$
– TheSimpliFire
Jul 19 at 14:49
I suppose you could reduce it to $$frac1x^2-frac1x^2(1+x^2)^n+1$$
– TheSimpliFire
Jul 19 at 14:49
@TheSimpliFire, yes, I did it. So do you think that there is no another way to write the sum using this formula?
– E. Shcherbo
Jul 19 at 14:51
@TheSimpliFire, yes, I did it. So do you think that there is no another way to write the sum using this formula?
– E. Shcherbo
Jul 19 at 14:51
No, I don't think so. The expression looks fine by itself. Why do you want such a simplification anyway?
– TheSimpliFire
Jul 19 at 14:53
No, I don't think so. The expression looks fine by itself. Why do you want such a simplification anyway?
– TheSimpliFire
Jul 19 at 14:53
@TheSimpliFire, When I haven't an answer a simple solution says to me that I solved it correctly :-)
– E. Shcherbo
Jul 19 at 14:55
@TheSimpliFire, When I haven't an answer a simple solution says to me that I solved it correctly :-)
– E. Shcherbo
Jul 19 at 14:55
This works for $xneq 0$
– Rumpelstiltskin
Jul 19 at 14:56
This works for $xneq 0$
– Rumpelstiltskin
Jul 19 at 14:56
 |Â
show 5 more comments
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1
I suppose you could reduce it to $$frac1x^2-frac1x^2(1+x^2)^n+1$$
– TheSimpliFire
Jul 19 at 14:49
@TheSimpliFire, yes, I did it. So do you think that there is no another way to write the sum using this formula?
– E. Shcherbo
Jul 19 at 14:51
No, I don't think so. The expression looks fine by itself. Why do you want such a simplification anyway?
– TheSimpliFire
Jul 19 at 14:53
@TheSimpliFire, When I haven't an answer a simple solution says to me that I solved it correctly :-)
– E. Shcherbo
Jul 19 at 14:55
This works for $xneq 0$
– Rumpelstiltskin
Jul 19 at 14:56