Question

Let \(0

Short answer

The series converges for real numbers \( p < 0 \).

Step-by-step solution

Understand the Series

The series in question is \( \sum_{n=1}^{\infty} n^{p} r^{n} \), where \( 0 < r < 1 \). This is an infinite series of the form \( n^{p} r^{n} \). We must find the values of \( p \) for which this series converges.

Use the Root Test for Convergence

For an infinite series \( \sum a_n \), the Root Test involves examining \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \). If \( L < 1 \), the series converges. In this scenario, \( a_n = n^p r^n \). Thus,\[L = \lim_{n \to \infty} \sqrt[n]{|n^p r^n|} = \lim_{n \to \infty} (n^p)^{1/n} r = \lim_{n \to \infty} n^{p/n} \cdot r.\]Since \( \lim_{n \to \infty} n^{p/n} = 1 \), \( L = r < 1 \) naturally since \( r < 1 \). The Root Test is inconclusive here because it only confirms \( L \leq r < 1 \) without additional conditions.

Apply the Comparison Test

To determine convergence, we compare our given series with a known convergent series. Consider the geometric series \( \sum_{n=1}^{\infty} r^{n} \), which is convergent since \( r < 1 \). The aim is to show \( n^p r^n \) behaves similarly to convergent terms when \( p < 0 \). Since \( n^p \) will tend to 0 as \( n \) becomes large when \( p < 0 \), \( n^p r^n \) mimics the convergence behavior of \( r^n \). Per comparison test logic, if \( 0 < r < 1 \), the given series converges for \( p < 0 \), since the terms become smaller than those of the geometric series.

Consider the Case of Divergence

To ensure the result, observe that for \( p \geq 0 \), the terms \( n^p r^n \) do not shrink sufficiently quickly to zero since \( n^p \) grows rapidly. For this reason, the series diverges if \( p \geq 0 \). Particularly for \( p = 0 \), the series becomes analogous to \( \sum r^n \) without the decay, and for \( p > 0 \), the growth of \( n^p \) ensures terms do not vanish at infinity, causing divergence.

Key concepts

Root Test

The Root Test is a method used to assess the convergence of an infinite series. Consider a series \( \sum a_n \). The test calculates the limit \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \).
Here's what each possible outcome indicates:

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \) or \( L = \infty \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

When applying the Root Test to the series \( \sum_{n=1}^{\infty} n^{p} r^{n} \), we find
that \( L = r \), as shown by the calculation in the solution.Given \( r < 1 \), this seems promising for convergence. However, since \( L \) only confirmsit is less than 1 without further narrowing down conditions, we find that the Root Test
isn't decisive here. Its inconclusiveness urges examination through other means, like the Comparison Test.

Comparison Test

The Comparison Test is a valuable tool when the Root Test is inconclusive, as it comparesterms of a given series to those of another series whose behavior we know. Suppose we have
\( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) as two series.

• If \( 0 \leq a_n \leq b_n \) and \( \sum b_n \) converges, then \( \sum a_n \) converges.
• If \( 0 \leq b_n \leq a_n \) and \( \sum b_n \) diverges, then \( \sum a_n \) diverges.

Considering the series \( \sum_{n=1}^{\infty} r^{n} \), which is a geometric series converging since
\( r < 1 \), we compare it with our original series term \( n^p r^n \).
If \( p < 0 \), \( n^p \to 0 \) as \( n \to \infty \), allowing \( n^p r^n \) to behave similarly to
\( r^n \), ensuring convergence. Thus, for \( p < 0 \), our series converges by the Comparison Test.

Infinite Series

Infinite series represent the summation of an infinite sequence of terms, providing a way to understand complex mathematical structures and behaviors. A series can either converge or diverge.

• Convergence: A series converges if it approaches a specific value as more terms are added.
• Divergence: A series diverges if, as terms are summed, it fails to settle to any limit.

In our problem, the series \( \sum_{n=1}^{\infty} n^{p} r^{n} \) has a specified range for \( r \), where \( 0 < r < 1 \), which influences its potential to converge.
When a series converges or diverges depends on the interplay between the terms. For this series:

• For \( p < 0 \), the series converges as terms diminish quickly.
• For \( p \geq 0 \), it diverges since terms do not decrease rapidly enough.

Understanding these characteristics helps solve complex convergence problems in calculus.