91Ó°ÊÓ

Monotonic sequences are a fundamental concept in sequences and series. A sequence is considered monotonic if it consistently increases or decreases. In our problem, the sequence \( S_n \) is defined by adding each new term \( a_{n+1} \). This means:

• For each \( n \), the sequence increases since the new term \( a_{n+1} > 0 \).
• Mathematically, we express that the monotonically increasing nature of \( S_n \) as \( S_{n+1} = S_n + a_{n+1} = S_n \left(1 + \frac{1}{n+1}\right) \) which confirms the increase as \( \left(1 + \frac{1}{n+1}\right) > 1 \).

Understanding monotonic behavior is crucial because it helps predict the sequence's growth direction and its bounds. For example, since \( S_n \) is always growing, we expect it to push the terms \( a_n \) towards zero, encouraging certain convergence properties.

A harmonic series is a specific type of series that can help understand other series' behaviors. It has the form \( \sum \frac{1}{n} \) and is famously known for its divergence.
In our exercise, there's an analogy to the harmonic series. As we explored the series of \( a_n \), we initially consider if our given series \( \sum a_n \) could behave like a convergent harmonic-like series. Unlike the traditional harmonic series whose terms approach zero too slowly to sum to a finite value, our sequence's definition constrains \( a_n \) to shrink faster.
The role that the harmonic series concept plays here is primarily in comparison. By considering how quickly \( a_n \) declines in comparison to terms of \( \frac{1}{n+1} \), we gain insight into the likelihood of convergence.

The comparison test is a powerful tool for series convergence questions. It's used to determine if a given series converges or diverges by comparing it to another series whose behavior we confidently know.

• The basic idea is: if our series \( a_n \) is compared with a known convergent series \( b_n \) and \( a_n \leq C b_n \) for some constant \( C \), then \( \sum a_n \) also converges.
• In our example, we see \( a_{n+1} \approx \frac{S_n}{n+1} \approx \frac{n}{n+1} \). But, unlike a simple harmonic series, our unique structure with \( S_n \approx n \) and rapid decrease in \( a_{n+1} \) after each term share resemblance to a modified convergent series.

By applying the comparison test, we conclude that since \( a_{n+1} \) fits within the bounds of a convergent series \( \frac{1}{n+1} \), the sum \( \sum_{n=1}^{\infty} a_n \) indeed converges. This aligns with our earlier hypothesis that although \( a_n \) declines similarly to a harmonic series, it does so more efficiently, enough to ensure convergence.