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

Guest Jun 6, 2020

#2**0 **

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**

Guest Jun 6, 2020