Question

What is the expression for the sum\[1+3x+5x^{2}+7x^{3}+9x^{4}....\]

Answer 1

there is an expression,but only if we are given x is a fraction..is it in this case ?

Answer 2

is x<1 ?

Answer 3

there is no criteria

Answer 4

then this cant be simplified to some expression i'd guess..
you see,,the derivation of the expression assumed x<1..

Answer 5

ok let x < 1

Answer 6

then the expression of sum is
ab/(1-r) + (dbr)/( (1-r)^2 )

a = 1st term of AP
b= 1st term of GP
d & r have there usual meanings..

Answer 7

this is arithmetico geometric series..................

Answer 8

the above thing has a very simple yet interesting derivation/proof..

Answer 9

1+3x+5x^2+7x^3 +9x^4...
=1+x+2x+x^2+4x^2 +x^3+6x^3+x^4+8x^4+...
=(1+x+x^2+x^3+x^4+...)+(2x+4x^2+6x^3+8x^4+...)
=(x^0+x^1+x^2+x^4+...)+2(x+2x^2+3x^3+4x^4+...)

Answer 10

That may lead to the solutiom

Answer 11

I got this answer\[\frac{ 3-x-2x^{n} }{ (x-1)^{2} }\] is it right?

Answer 12

Thanks