Compute the value of x such that\(\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}\cdots\right)\left(1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots\)h
The first sequence sums up to =3/2
The second sequence sums up to =3/4
3/2 x 3/4 = 9/8
These are closed form formulas to sum up the two sequences:
(1 + 1/3 + 1/9 + 1/27 + ...) = sum_(n=1)^∞ 3^(1 - n) = 3/2
(1 - 1/3 + 1/9 - 1/27 + ...) = sum_(n=1)^∞ (-1)^(1 + n) 3^(1 - n) = 3/4
The RHS of 1 + 1/x + 1/x^2 + 1/x^3 + 1/x^4 +.... = x /(x - 1)
Solve for x:
x/(x - 1) = 9/8
Multiply both sides by a polynomial to clear fractions.
Cross multiply:
8 x = 9 (x - 1)
Write the linear polynomial on the right hand side in standard form.
Expand out terms of the right hand side:
8 x = 9 x - 9
Isolate x to the left hand side.
Subtract 9 x from both sides:
-x = -9
Solve for x.
Multiply both sides by -1:
x = 9
