91Ó°ÊÓ

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.

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}, ...

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.