Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k.

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} k \) is the sum of natural numbers starting from 1.

Understand that the sequence of partial sums \( S_n \) is defined as \( S_n = \sum_{k=1}^n k \), which means adding the first \( n \) terms of the series.

Calculate the first partial sum \( S_1 \) by summing the first term: \( S_1 = 1 \).

Calculate the second partial sum \( S_2 \) by summing the first two terms: \( S_2 = 1 + 2 \).

Calculate the third and fourth partial sums similarly: \( S_3 = 1 + 2 + 3 \) and \( S_4 = 1 + 2 + 3 + 4 \).

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주요 개념

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Infinite Series and Partial Sums

An infinite series is the sum of infinitely many terms. The sequence of partial sums, Sₙ, represents the sum of the first n terms of the series. Evaluating partial sums helps understand the behavior of the series, especially whether it converges or diverges.

Arithmetic Series

An arithmetic series is a sum of terms with a constant difference between consecutive terms. For the series ∑k, the terms increase by 1 each time. The sum of the first n terms can be found using the formula Sₙ = n(n + 1)/2.

Summation Notation and Indexing

Summation notation (∑) compactly represents the sum of terms indexed by k. Understanding how to interpret and manipulate the index and limits is essential for correctly evaluating partial sums and applying formulas.