Use the properties of geometric series to find the sum of the given series. For what value of the variable does the series converge to this sum? 1+z/6+z^2/36+z^3/216+⋯ Enclose numerators, and denominators in parentheses. For example, (a−b)/(1+n). The sum of the given series is:

for

Step-by-step explanation:

Given

Required

The value at which the series converges

Calculate r

For a series to converge:

This gives:

Multiply both sides by 6

This is calculated using the sum to infinity of a gp.

Where

So:

Take LCM

Rewrite as:

So, the function converges at:

for

27 +27 or 27times 2=54

the diffrence is 0