Question

A series is given. (a) Find a formula for \(S_{n},\) the \(n^{\text {th }}\) partial sum of the series. (b) Determine whether the series converges or diverges. If it converges, state what it converges to. $$1^{3}+2^{3}+3^{3}+4^{3}+\cdots$$

Short answer

The series diverges since the sum approaches infinity as \(n\) increases.

Step-by-step solution

Identify the Type of Series

The given series is a sum of cubes: \[1^3 + 2^3 + 3^3 + 4^3 + \cdots\]This is a series formed by summing the cubes of natural numbers. This will help identify a formula for the nth partial sum.

Recall the Formula for the Sum of Cubes

The formula for the sum of the first \(n\) cubes is known: \[S_n = (1 + 2 + 3 + \cdots + n)^2\]Using the formula for the sum of the first \(n\) natural numbers, \(\frac{n(n+1)}{2}\), the nth partial sum can be written as: \[S_n = \left(\frac{n(n+1)}{2}\right)^2\].

Write the Formula for the Partial Sum

Based on Step 2, the formula for the nth partial sum is:\[S_n = \left(\frac{n(n+1)}{2}\right)^2\].

Analyze Convergence or Divergence

The given series sums the cubes of natural numbers indefinitely as \(n\) tends towards infinity.

Determine Divergence or Convergence

Since the cubes of natural numbers grow without bound and their sum \((1 + 2 + 3 + \cdots + n)^2\) also grows without bound, the series diverges. This can be concluded from the fact that the sum approaches infinity as \(n\) approaches infinity.

Key concepts

Partial Sum Formula

When dealing with series, especially those involving sums of powers, a partial sum formula is crucial. The partial sum for a series represents the sum of the first \(n\) terms. For the series consisting of cubes, each term is written as the cube of a consecutive natural number, resulting in a sequence like \(1^3, 2^3, 3^3, \ldots, n^3\). This partial sum formula is important because it helps simplify the calculation of the series up to a finite number of terms.In our specific scenario, the formula for the sum of cubes is derived from the identity for the sum of the first \(n\) natural numbers. Using this identity, the partial sum formula for the cubes becomes:

• First, recall the sum of first \(n\) natural numbers: \(\frac{n(n+1)}{2}\)
• Then, the nth partial sum formula for cubes is \(S_n = \left(\frac{n(n+1)}{2}\right)^2\).

By calculating this expression for any \(n\), we can find the sum of the cubes up to that number.

Sum of Cubes

The sum of cubes is a particular series where each term is the cube of a natural number. For example, \(1^3 + 2^3 + 3^3 + \cdots\). Unlike simple arithmetic series where each term increases by a constant, in the sum of cubes, the terms increase more rapidly because they are raised to the third power.The sum of cubes has an interesting mathematical property whereby it can be expressed as the square of the sum of natural numbers. This remarkable property gives rise to the formula:

• All cubes summed: \(1^3 + 2^3 + 3^3 + \cdots + n^3\)
• Is equivalent to the square: \((1 + 2 + 3 + \cdots + n)^2\)

Thus, the sum of cubes up to any given number \(n\) is simply the square of the sum of the numbers up to \(n\). This geometric aspect of cubes summing to a perfect square makes them a unique study in mathematics.

Infinite Series

In mathematics, an infinite series is a sum of infinite terms. It is often denoted by a sequence that continues indefinitely. For example, the series \(1^3 + 2^3 + 3^3 + 4^3 + \cdots\) is an infinite series.Infinite series can be both intriguing and complicated because they require a deeper understanding of summation past a finite limit. Exploring an infinite series involves identifying whether it approaches a finite value or not. This quest relates closely to concepts of convergence and divergence.The series of cubes is a great example. While there is a formula for summing the cubes up to any specific \(n\), asking about the sum from \(n=1\) to infinity tests whether the series converges to a finite number or diverges to infinity. Due to the nature of cubed terms growing quickly, it is recognized that such series usually prove divergent when extended to infinity.

Convergence and Divergence

One of the key questions in studying series, especially infinite series, is whether they converge or diverge. Convergence implies that as we sum more terms, the total approaches a specific finite value. In contrast, divergence suggests that the sum grows without limit.Determining convergence or divergence can depend on multiple factors, such as the type of series and the behavior of its terms. For the infinite series \(1^3 + 2^3 + 3^3 + \cdots\), we analyze it by considering the growth rate of its terms:

• Each term \(n^3\) tends to become very large as \(n\) increases.
• The summation grows without bounds because the cube function increases rapidly.

Therefore, we classify this series as divergent, meaning it does not settle to any finite value as we consider more terms. Recognizing whether a series converges or diverges allows us to understand its limitations and potential applications in mathematical calculations.