Question

Determine whether the following statements are true and give an explanation or counterexample. a. The sequence of partial sums for the series \(1+2+3+\cdots\) is \(\\{1,3,6,10, \ldots\\}\) b. If a sequence of positive numbers converges, then the terms of the sequence must decrease in size. c. If the terms of the sequence \(\left\\{a_{n}\right\\}\) are positive and increasing, then the sequence of partial sums for the series \(\sum_{k=1}^{2} a_{k}\) diverges.

Short answer

Answer: Yes. The sequence of partial sums matches the given sequence, \(\{1, 3, 6, 10, \ldots\}\). b. Is the statement "if a sequence of positive numbers converges, then its terms must decrease in size" true? Answer: No. The statement is false, as demonstrated by the counterexample of the sequence \(\left\\{a_n\right\\} = \left\\{\frac{1}{n}+\frac{1}{n^2}\right\\}\), which converges but does not have decreasing terms. c. Does the corrected statement "if the terms of the sequence \(\left\\{a_n\right\\}\) are positive and increasing, then the sequence of partial sums for the series \(\sum_{k=1}^{\infty} a_{k}\) diverges" hold true? Answer: Yes. The corrected statement is true, as a positive, non-decreasing series will have strictly increasing partial sums, and thus will not converge (it will diverge).

Step-by-step solution

Find the sequence of partial sums

The sequence of partial sums is \(\{S_n\}\) where \(S_n = \sum_{k=1}^n k\). In other words, \(S_1 = 1\), \(S_2 = 1+2\), \(S_3 = 1+2+3\), and so on. We can continue this sequence for the first few terms to verify if it matches the given one.

Verify the sequence of partial sums

The sequence of partial sums is \(\\{1, 1+2, 1+2+3, 1+2+3+4, \ldots \\}=\\{1, 3, 6, 10, \ldots\\}\). This matches the given partial sums, so the statement is true. #Statement b#

Evaluate the statement

The statement says that if a sequence of positive numbers converges, then the terms must decrease in size. This might seem reasonable at first glance, but it's essential to verify whether it is true or provide a counterexample.

Find a counterexample

Consider the sequence \(\left\\{a_n\right\\} = \left\\{\frac{1}{n}+\frac{1}{n^2}\right\\}\). All terms are positive, and \(\lim_{n \to \infty} \frac{1}{n}+\frac{1}{n^2}=0\). However, the terms of the sequence are not decreasing in size; for example, \(a_1=\frac{3}{2}\) while \(a_2=\frac{5}{4}>\frac{3}{2}\). Hence, the statement is false. #Statement c#

Evaluate the statement

The statement says that if the terms of the sequence \(\left\\{a_n\right\\}\) are positive and increasing, then the sequence of partial sums for the series \(\sum_{k=1}^{2} a_{k}\) diverges. We need to determine whether this is true or provide a counterexample.

Correct the series

The statement has an error in the series notation; it should be \(\sum_{k=1}^{\infty} a_{k}\), not \(\sum_{k=1}^{2} a_{k}\). With this correction, the statement would be: "if the terms of the sequence \(\left\\{a_n\right\\}\) are positive and increasing, then the sequence of partial sums for the series \(\sum_{k=1}^{\infty} a_{k}\) diverges."

Determine the truth of the corrected statement

Since the sequence \(\left\\{a_n\right\\}\) is positive and increasing, the series \(\sum_{k=1}^{\infty} a_{k}\) is a positive, non-decreasing series. Therefore, the sequence of partial sums is strictly increasing, and there is no real number that the partial sums approach as \(n\) goes to infinity. In other words, the sequence of partial sums does not converge (and therefore diverges). The corrected statement is true.

Key concepts

Partial Sums

A partial sum refers to the sum of the first few terms in a sequence. Think of it as a running total as you add up each term, one by one. In the context of a series such as 1+2+3+... , the partial sums are the sums obtained at each step when you add each term of the sequence:

• For the first term: \(S_1 = 1\)
• For the first two terms: \(S_2 = 1 + 2 = 3\)
• For the first three terms: \(S_3 = 1 + 2 + 3 = 6\)

These values form the sequence \( \{1, 3, 6, 10, \ldots\} \), showing how the sum increases as more terms are added. Each partial sum simply captures where the sum stands partway through the infinite process of adding all items in the series.

Divergence and Convergence

In the study of series, understanding whether a series converges or diverges is crucial. A series converges if its partial sums approach a specific number as you add more terms, no matter how many. Conversely, it diverges if the sums continue to grow larger indefinitely or oscillate without reaching a specific limit.
When we say a series of positive, increasing terms will diverge, this stems from the idea that adding bigger and bigger numbers should keep the sum growing without bound. Essentially, there’s no real number to "converge" to.

• Converges: Sum stops growing at a value.
• Diverges: Sum keeps growing without end.

Recognizing this helps in determining if a given series has a manageable size or endless growth.

Counterexamples

A counterexample is a specific case for which a general statement is false. For example, a common error is assuming that any convergent sequence must consist of decreasing terms. However, counterexamples can be provided to disprove this misconception.
The sequence defined by \(a_n = \frac{1}{n} + \frac{1}{n^2}\) converges to zero although its terms do not decrease steadily. Initially, the terms increase, such as when \(a_1 = \frac{3}{2}\) and \(a_2 = \frac{5}{4} > \frac{3}{2}\). This example shows it is not necessary for terms to decrease in magnitude to achieve convergence. Thus, counterexamples are powerful tools that highlight exceptions to erroneous rules or intuitions.