Question

Find the sum of the convergent series. $$ \sum_{n=0}^{\infty} 2\left(-\frac{2}{3}\right)^{n} $$

Short answer

The sum of the series \(\sum_{n=0}^{\infty} 2(-2/3)^{n}\) is 1.2

Step-by-step solution

Identify the values of a and r

Firstly, analyze the series. Here, the first term (\(a\)) is 2 and the common ratio (\(r\)) is \(-2/3\)

Substitute the values into the sum formula

The next step is to substitute the values of \(a\) and \(r\) into the sum formula of an infinite geometric series. The formula is \(S = \frac{a}{1 - r}\). So the sum \(S\) of the series is \(S = \frac{2}{1 - (-2/3)}\)

Calculate the sum

Solve the equation \(S = \frac{2}{1 - (-2/3)}\). Multiply the numerator and the denominator by 3 to get rid of the fraction in the denominator: \(S = \frac{2*3}{3 - (-2)} = \frac{6}{3 + 2} = 6/5=1.2\)