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Chapter 9: Problem 7

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

Short Answer

Expert verified
Answer: The sequence of partial sums of a series with positive terms is an increasing sequence because each new term added to the sum is positive. This means that each subsequent partial sum is greater than the previous one.

Step by step solution

01

Define a sequence with positive terms

Let \(a_n\) be a sequence of positive terms, where \(a_n > 0\) for each term, and let \(n \in \mathbb{N}\).
02

Define the partial sums

For a given sequence \(a_n\), we define the partial sums \(S_n\) as follows: \(S_k = \sum_{n=1}^k a_n\) for \(k \in \mathbb{N}\). This means that \(S_1 = a_1\), \(S_2 = a_1 + a_2\), \(S_3 = a_1 + a_2 + a_3\), and so on.
03

Show that the sequence of partial sums is increasing

To prove that the sequence of partial sums is increasing, we need to show that for any two consecutive partial sums, \(S_k\) and \(S_{k+1}\), we have \(S_{k+1} > S_k\). From the definition of partial sums, we know that \(S_{k+1} = S_k + a_{k+1}\). Since \(a_{k+1}>0\), it follows that \(S_{k+1} > S_k\), proving that the sequence of partial sums is increasing. Therefore, with a series of positive terms, the sequence of partial sums is an increasing sequence.

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