Problem 5 - Essential Calculus

Question

Find the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty} \frac{3}{2^{n-1}} $$

Step-by-Step Solution

Answer

Therefore, the first five terms of the sequence of partial sums are: 3, 4.5, 5.25, 5.625, 5.8125.

Step 1: Identify the first term and the common ratio

In the series \( \sum_{n=1}^{\infty} \frac{3}{2^{n-1}} \), the first term 'a' is 3. The common ratio 'r', which is the factor between any two consecutive terms, is \(0.5\)

Step 2: Calculate the first term of the partial sums

In this case, the first term of the partial sums is equal to the first term of the series since it has no earlier terms to add. So, the first term of the partial sums is 3.

Step 3: Calculate the second term of the partial sums

The second term of the partial sums is the sum up to the second term in the series. By substituting \(n = 2\), \(a = 3\) and \(r = 0.5\) into \(S_n = a \frac{1-r^n}{1-r}\), the result is 3 + 1.5 = 4.5.

Step 4: Calculate the third term of the partial sums

The third term of the partial sums is the sum up to the third term in the series. By substituting \(n = 3\), \(a = 3\) and \(r = 0.5\) into \(S_n = a \frac{1-r^n}{1-r}\), the result is 4.5 + 0.75 = 5.25.

Step 5: Calculate the fourth term of the partial sums

The fourth term of the partial sums is the sum up to the fourth term in the series. By substituting \(n = 4\), \(a = 3\) and \(r = 0.5\) into \(S_n = a \frac{1-r^n}{1-r}\), the result is 5.25 + 0.375 = 5.625.

Step 6: Calculate the fifth term of the partial sums

The fifth term of the partial sums is the sum up to the fifth term in the series. By substituting \(n = 5\), \(a = 3\) and \(r = 0.5\) into \(S_n = a \frac{1-r^n}{1-r}\), the result is 5.625 + 0.1875 = 5.8125.