Question

Find the sum of the convergent series. $$ 4-2+1-\frac{1}{2}+\cdots $$

Short answer

The sum of the series is \(\frac{8}{3}\).

Step-by-step solution

Identify the values

First, identify the first term and the ratio of the series. Here, the first term \(a\) is 4 and the ratio \(r\) is \(-\frac{1}{2}\).

Apply the geometric series sum formula

Next, apply the formula for the sum of a geometric series, which is \(S = \frac{a}{1-r}\). Substituting in the known values, we get \(S = \frac{4}{1 - (-\frac{1}{2})}\)

Calculate the sum

Now, compute this to get the sum. Doing so gives \(S = \frac{4}{1+\frac{1}{2}} = \frac{4}{\frac{3}{2}} = \frac{4*2}{3} = \frac{8}{3}.\)