Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.

c.If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series∑⁽∞⁾ₖ₌₁aₖ diverges.

검증된 단계별 안내

Recall that the sequence {aₙ} consists of positive and increasing terms, meaning \( aₙ > 0 \) and \( a_{n+1} \geq a_n \) for all \( n \).

The series in question is \( \sum_{k=1}^{\infty} a_k \), and its partial sums are defined as \( S_n = \sum_{k=1}^n a_k \).

Since each term \( a_k \) is positive, the partial sums \( S_n \) form an increasing sequence because adding a positive term increases the sum: \( S_{n+1} = S_n + a_{n+1} > S_n \).

Because the terms \( a_n \) are increasing and positive, the terms do not approach zero. In fact, \( \lim_{n \to \infty} a_n \neq 0 \), which is a necessary condition for series convergence.

Therefore, since the terms do not tend to zero, the series \( \sum_{k=1}^{\infty} a_k \) must diverge, meaning the sequence of partial sums \( S_n \) diverges to infinity.

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Monotonic Sequences

A sequence is monotonic if it is either entirely non-increasing or non-decreasing. In this question, the sequence {aₙ} is positive and increasing, meaning each term is greater than or equal to the previous one. Understanding monotonicity helps analyze the behavior and limits of sequences.

Partial Sums and Series Convergence

The partial sums of a series are the sums of its first n terms. The convergence or divergence of a series depends on whether these partial sums approach a finite limit as n approaches infinity. If the partial sums grow without bound, the series diverges.

Comparison Test and Counterexamples in Series

To determine if a series converges or diverges, comparison tests can be used by comparing with known series. Counterexamples are important to disprove general statements; for instance, an increasing positive sequence might have partial sums that diverge or converge, depending on the terms' growth rate.

Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.c.If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series∑⁽∞⁾ₖ₌₁aₖ diverges.