Question

Given that the following two geometric series(meetkundige reekse) are covergent 1+x+x²+x³+... And 1-x+x²-x³+... Determine the value(s) of x for witch the sum of the following two series is equel​

Answer 1

The sets are (1,-1) or (-1,1)

Lets see

Take until 4terms

#1

$\\ \rm\longmapsto S_4=1+1+1^2+1^3=1+1+1+1=4$(1,-1)

$\\ \rm\longmapsto S_4=1+(-1)+(-1)^2+(-1)^3=1-1+1-1=0$(-1,1)

#2

$\\ \rm\longmapsto S_4=1-(-1)+(-1)^2-(-1)^3=1+1+1+1=4$(1,-1)

$\\ \rm\longmapsto S_4=1-1+(1)^2-1^3=1-1+1-1=0$(-1,1)

Hence verified

Another set can be (0,0)

Proof:-

$\\ \ast\rm\longmapsto S_3=1+0+0^2=1$

$\\ \ast\rm\longmapsto S_3=1-0+0^2=1$

Hence verified.