Use a geometric series to represent....

Question

Use a geometric series to represent 1/(3−2x) as a power series about x = 0, and find its intervals of convergence.

Answer 1

a/(1 - r) is the sum of a geometric series with first term a and ratio r. The sum is a + ar + ar² + ar³ ...

We may put 1/(3 - 2x) in this form

Let's multiply the numerator and denominator of 1/(3 - 2x) by 1/3/(1/3)

Then, we have

1/3/(1 - 2/3 x)

Now, we have this in the form of the geometric series with first term 1/3 and ratio 2/3 x

The series is ∑ 1/3(2/3 x)ⁿ as n goes from 0 to infinity

This series converges when the absolute value of the ratio, |2/3 x| < 1 or

|x| < 3/2 or the interval is (-3/2, 3/2)