Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.

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Recall the definitions: A sequence \( \{a_k\} \) converges if \( \lim_{k \to \infty} a_k = L \) for some finite number \( L \). A series \( \sum_{k=1}^\infty a_k \) converges if the sequence of partial sums \( S_n = \sum_{k=1}^n a_k \) converges to a finite limit.

Note that if the series \( \sum a_k \) converges, then the terms \( a_k \) must approach zero. However, the converse is not true: \( a_k \to 0 \) does not guarantee that \( \sum a_k \) converges.

To find an example where \( \{a_k\} \) converges but \( \sum a_k \) diverges, consider the sequence \( a_k = \frac{1}{k} \). This sequence converges to zero as \( k \to \infty \).

Examine the series \( \sum_{k=1}^\infty \frac{1}{k} \), known as the harmonic series. It is a classic example of a divergent series despite its terms tending to zero.

Thus, the sequence \( a_k = \frac{1}{k} \) converges to zero, but the series \( \sum_{k=1}^\infty a_k = \sum_{k=1}^\infty \frac{1}{k} \) diverges.

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주요 개념

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Convergence of a Sequence

A sequence {aₖ} converges if its terms approach a specific finite limit as k approaches infinity. This means for large k, the terms get arbitrarily close to that limit. For example, the sequence aₖ = 1/k converges to 0.

Divergence of a Series

A series ∑ aₖ diverges if the sum of its terms does not approach a finite limit as the number of terms grows. Even if the terms aₖ approach zero, the series can still diverge, such as the harmonic series ∑ 1/k.

Relationship Between Sequence and Series Convergence

While a sequence {aₖ} may converge to zero, the corresponding series ∑ aₖ might diverge. This distinction is crucial because term-wise convergence to zero is necessary but not sufficient for series convergence.