Problem 3 - Essential Calculus

Question

Find the first five terms of the sequence of partial sums. $$ 3-\frac{9}{2}+\frac{27}{4}-\frac{81}{8}+\frac{243}{16}-\cdots $$

Step-by-Step Solution

Answer

The first five terms of the sequence of partial sums are: \(3, -\frac{3}{2}, \frac{9}{4}, -\frac{9}{8}, \frac{135}{32}\)

Step 1: Note down the Terms

We note down the first 5 terms of the series as given in the exercise, these are \(3, -\frac{9}{2}, \frac{27}{4}, -\frac{81}{8}, \frac{243}{16} \)

Step 2: Calculate Partial Sums

We next calculate the partial sums (the sum of the series up to a certain number of terms) by adding each new term to the sum of previous terms. We denote the partial sum by \(S_n\).

Step 3: Find First Partial Sum (S1)

The first partial sum \(S_1\) is just the first term of the series, which is \(3\).

Step 4: Find Second Partial Sum (S2)

The second partial sum \(S_2\) is the sum of the first and second term. It is \( S_2 = 3 -( \frac{9}{2}) = -\frac{3}{2}\).

Step 5: Find Third Partial Sum (S3)

The third partial sum \( S_3 \) is found by adding the third term to the second partial sum, it is \(S_3 = -\frac{3}{2} + \frac{27}{4} = \frac{9}{4}\).

Step 6: Find Fourth Partial Sum (S4)

The fourth partial sum \( S_4 \) is found by adding the fourth term to the third partial sum, it is \( S_4= \frac{9}{4} - \frac{81}{8} = -\frac{9}{8}\).

Step 7: Find Fifth Partial Sum (S5)

The fifth partial sum \( S_5 \) is found by adding the fifth term to the fourth partial sum, it is \( S_5 = -\frac{9}{8} + \frac{243}{16} = \frac{135}{32}\).