Problem 8 - Calculus Early Transcendentals

Question

Evaluate each geometric sum. $$\sum_{k=0}^{10}\left(\frac{1}{4}\right)^{k}$$

Step-by-Step Solution

Answer

Question: Calculate the value of the following geometric sum: $$\sum_{k=0}^{10}\left(\frac{1}{4}\right)^{k}$$ Answer: The value of the geometric sum is approximately 1.333225.

Step 1: Identify the first term (a_1)

The first term \(a_1\) is the value that the sum starts with. In this case, it is the term with \(k=0\), which results in \(\left(\frac{1}{4}\right)^0 = 1\). So \(a_1 = 1\).

Step 2: Identify the common ratio (r)

The common ratio \(r\) is the factor by which the terms in the geometric sequence are multiplied. In this case, since the sum is given as a power of \(\frac{1}{4}\), the common ratio is \(\frac{1}{4}\).

Step 3: Identify the number of terms (n)

The number of terms to sum is given by \(k=0\) to \(k=10\). Since the index starts at 0, there are eleven terms in total. So \(n = 11\).

Step 4: Apply the geometric sum formula

Plugging the values of \(a_1\), \(r\), and \(n\) into the formula, we get: $$S_{11} = \frac{1\left(1-\left(\frac{1}{4}\right)^{11}\right)}{1-\frac{1}{4}}$$

Step 5: Simplify the expression

Simplify the expression by performing the operations: $$S_{11} = \frac{1\left(1-\left(\frac{1}{4}\right)^{11}\right)}{\frac{3}{4}}$$

Step 6: Calculate the sum

Now we calculate the sum: $$S_{11} = \frac{4\left(1-\left(\frac{1}{4}\right)^{11}\right)}{3} \approx 1.333225$$ So the value of the geometric sum $$\sum_{k=0}^{10}\left(\frac{1}{4}\right)^{k}$$ is approximately 1.333225.