Problem 22 - Thomas Calculus

Question

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{n+1}{n^{2} \sqrt{n}}\end{equation}

Step-by-Step Solution

Answer

The series converges because it is the sum of two convergent p-series.

Step 1: Simplify the General Term

The given series is \( \sum_{n=1}^{\infty} \frac{n+1}{n^{2} \sqrt{n}} \). Let's simplify the general term: \( \frac{n+1}{n^{2} \sqrt{n}} = \frac{n}{n^{2} \sqrt{n}} + \frac{1}{n^{2} \sqrt{n}} = \frac{1}{n^{3/2}} + \frac{1}{n^{5/2}} \).

Step 2: Test Each Simplified Series for Convergence

The simplified series consists of two series: 1. \( \sum \frac{1}{n^{3/2}} \) 2. \( \sum \frac{1}{n^{5/2}} \) For both series, since \( p > 1 \), they converge according to the p-series test.

Step 3: Conclude the Convergence of the Original Series

The original series is the sum of two convergent series: \( \sum \frac{1}{n^{3/2}} \) and \( \sum \frac{1}{n^{5/2}} \). Since both of these series converge, the original series, \( \sum_{n=1}^{\infty} \frac{n+1}{n^{2} \sqrt{n}} \), also converges.

Key Concepts

P-series test

The P-series test is a fundamental technique in determining series convergence, especially when dealing with expressions involving powers of n. A P-series is in the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where p is a constant real number. The convergence of a P-series is decided by the value of p:

• If \( p > 1 \), the series converges.
• If \( p \leq 1 \), the series diverges.

In the example at hand, the series was broken down into two simpler P-series. By applying the P-series test:

• For \( \sum \frac{1}{n^{3/2}} \), we find that \( p = \frac{3}{2} \), which indicates convergence because \( 3/2 > 1 \).
• Similarly, for \( \sum \frac{1}{n^{5/2}} \), \( p = \frac{5}{2} \) suggests convergence as \( 5/2 > 1 \).

This step is crucial as it helps conclude whether the complicated or intricate series will sum up to a finite number.

Convergent series

A series is deemed convergent if the sum of its terms approaches a specific finite value as more terms are added. This happens when the series doesn't continue indefinitely without bounds. In mathematical terms, convergence occurs if the sequence of partial sums,

• \( S_n = a_1 + a_2 + \cdots + a_n \)

reaches a definite limit. In our exercise, each of the individual series \( \sum \frac{1}{n^{3/2}} \) and \( \sum \frac{1}{n^{5/2}} \) was independently evaluated using the P-series test. Since both series converged individually, we deduce that when combined, the original series is also convergent.

• Mathematically, the sum of two convergent series is always convergent.

Understanding this concept helps in determining whether an original complex series derives a finite value, impacting calculations in various scientific fields.

Simplification of series

Simplifying a series is the pivotal step that leads to easier analysis and understanding of its convergence properties. Simplification involves breaking down a complex expression into simpler components, making it easier to apply convergence tests.

• In this exercise, the original series \( \sum_{n=1}^{\infty} \frac{n+1}{n^{2} \sqrt{n}} \) was initially too complex for direct application of the P-series test.

The simplification process entailed expressing it as \( \frac{1}{n^{3/2}} + \frac{1}{n^{5/2}} \), thereby breaking down the series into two well-recognized forms that allowed straightforward application of the P-series test.

• This not only made the series easier to handle mathematically but ensured that accurate conclusions about convergence could be drawn.

Thus, simplification is an essential skill in series analysis, enabling students to tailor their understanding and approach according to a series' unique characteristics.