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

a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

검증된 단계별 안내

Recall that the sequence of partial sums of a series is formed by adding the terms of the series one by one. For the series 1 + 2 + 3 + ⋯, the nth partial sum is given by \(S_n = 1 + 2 + 3 + \cdots + n\).

Use the formula for the sum of the first n natural numbers: \(S_n = \frac{n(n+1)}{2}\). This formula gives the value of the nth partial sum directly.

Calculate the first few partial sums using the formula: for n=1, \(S_1 = \frac{1 \times 2}{2} = 1\); for n=2, \(S_2 = \frac{2 \times 3}{2} = 3\); for n=3, \(S_3 = \frac{3 \times 4}{2} = 6\); for n=4, \(S_4 = \frac{4 \times 5}{2} = 10\).

Compare these values to the given sequence {1, 3, 6, 10, …}. Since they match exactly, the sequence of partial sums for the series 1 + 2 + 3 + ⋯ is indeed {1, 3, 6, 10, …}.

Therefore, the statement is true because the sequence of partial sums corresponds to the triangular numbers given by the formula \(S_n = \frac{n(n+1)}{2}\).

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주요 개념

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

Sequence of Partial Sums

A sequence of partial sums is formed by adding the terms of a series one by one. For the series 1 + 2 + 3 + ⋯, the nth partial sum is the sum of the first n natural numbers, which creates a new sequence representing cumulative totals.

Formula for the Sum of the First n Natural Numbers

The sum of the first n natural numbers is given by the formula n(n + 1)/2. This formula helps quickly find the nth partial sum without adding each term individually, confirming the sequence of partial sums as {1, 3, 6, 10, …}.

Convergence and Divergence of Series

A series converges if its sequence of partial sums approaches a finite limit; otherwise, it diverges. Since the partial sums 1, 3, 6, 10, … grow without bound, the series 1 + 2 + 3 + ⋯ diverges, meaning it does not sum to a finite value.

Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.