Problem 115 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the indicated partial sum of the series. \(\sum_{n=1}^{\infty} 4\left(-\frac{1}{2}\right)^{n}\) Third partial sum

Step-by-Step Solution

Answer

The third partial sum (S3) of the given geometric series is 4.66667 or approximately \( \frac{14}{3} \).

Step 1: Identify the series terms and common ratio

In this series, the first term 'a' is clearly 4 and the common ratio 'r' is \(-\frac{1}{2}\). So, we are trying to find the sum of the first three terms of the series 4, -2, and 1.

Step 2: Use partial sum formula

To find the sum of the first three terms (n=3), we use the formula for the sum of the first 'n' terms of a geometric series, which is \( S_n = a (1 - r^n) / (1 - r) \). Substituting the identified values into the formula gives us \( S_3 = 4 * (1 - (-\frac{1}{2})^3) / {1 - (-\frac{1}{2})} \).

Step 3: Solve for S3

Solving the equation gives us the third partial sum, S3. Perform the computations within the parentheses first, according to the order of operations (BIDMAS/BODMAS/PEDMAS rule), then proceed to the division. The result is the required third partial sum.