Determine whether the following statements are true and give an explanation or counterexample.
c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.

검증된 단계별 안내

Recall that the sequence of partial sums \( S_n = \sum_{k=1}^n a_k \) represents the sum of the first \( n \) terms of the series \( \sum a_k \).

If the terms of the sequence of partial sums \( S_n \) approach a limit \( L \) as \( n \to \infty \), then the infinite series \( \sum a_k \) converges to \( L \).

In this problem, it is given that the terms of the sequence of partial sums approach \( \frac{5}{2} \). This means \( \lim_{n \to \infty} S_n = \frac{5}{2} \).

Since the partial sums approach \( \frac{5}{2} \), by definition, the infinite series \( \sum a_k \) converges and its sum is \( \frac{5}{2} \).

Therefore, the statement is true because the convergence of the partial sums to \( \frac{5}{2} \) directly implies the series converges to \( \frac{5}{2} \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sequence of Partial Sums

The sequence of partial sums is formed by adding the first n terms of a series. It helps analyze the behavior of the series by examining the limit of these sums as n approaches infinity.

Convergence of an Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. This limit is the sum of the series, meaning the series adds up to a specific value.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence get arbitrarily close to as the index grows large. If the partial sums approach 5/2, the series converges to 5/2.

Determine whether the following statements are true and give an explanation or counterexample.c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.