Problem 7 - Calculus Early Transcendentals

Question

Explain why, with a series of positive terms, the sequence of partial sums is an increasing sequence.

Step-by-Step Solution

Answer

Answer: The sequence of partial sums for a series with positive terms is an increasing sequence because each new term added is positive, making the new partial sum greater than the previous one. As we progress through the series, each additional term increases the partial sum, creating an increasing sequence of partial sums.

Step 1: Understand what a sequence of partial sums is

A sequence of partial sums for a given series is when we sum up the terms of the series up to a certain point. For example, for the series S = a_1 + a_2 + a_3 + ...: - The first partial sum, S_1, is the sum of the first term: S_1 = a_1 - The second partial sum, S_2, is the sum of the first two terms: S_2 = a_1 + a_2 - The third partial sum, S_3, is the sum of the first three terms: S_3 = a_1 + a_2 + a_3 And so on.

Step 2: Provide examples of series with positive terms and their sequence of partial sums

Let's consider two series with positive terms: Example 1: The arithmetic series with positive terms: 2 + 4 + 6 + 8 + ... - The first partial sum, S_1 = 2 - The second partial sum, S_2 = 2 + 4 = 6 - The third partial sum, S_3 = 2 + 4 + 6 = 12 As we can see, the sequence of partial sums for this series is an increasing sequence: 2, 6, 12, ... Example 2: The geometric series with positive terms: 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... - The first partial sum, S_1 = 1 - The second partial sum, S_2 = 1 + \frac{1}{2} = \frac{3}{2} - The third partial sum, S_3 = 1 + \frac{1}{2} + \frac{1}{4} = \frac{7}{4} Again, we see that the sequence of partial sums is an increasing sequence: 1, \frac{3}{2}, \frac{7}{4}, ...

Step 3: Proving that the sequence of partial sums for a series with positive terms is an increasing sequence

To show that the sequence of partial sums for a series with positive terms is an increasing sequence, we must prove that for all n, S_n+1 > S_n where S_n is the nth partial sum. If a series has only positive terms, then for any n, a_n > 0. Now consider S_n and S_{n+1}: S_n = a_1 + a_2 + ... + a_n S_{n+1} = a_1 + a_2 + ... + a_n + a_{n+1} = S_n + a_{n+1} Since a_{n+1} > 0 (as the terms are all positive), S_{n+1} must be greater than S_n: S_{n+1} = S_n + a_{n+1} > S_n And since this holds for any n, we can conclude that for a series with only positive terms, the sequence of partial sums is an increasing sequence.