Question

For each of the following series, use the root test to determine whether the series converges or diverges. a. \(\sum_{n=1}^{\infty} \frac{\left(n^{2}+3 n\right)^{n}}{\left(4 n^{2}+5\right)^{n}}\) b. \(\sum_{n=1}^{\infty} \frac{n^{n}}{(\ln (n))^{n}}\)

Short answer

a. Converges; b. Diverges.

Step-by-step solution

Define the Series

For part a, the series is given by \( \sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} \frac{(n^2+3n)^n}{(4n^2+5)^n} \). We need to find \( a_n \) and apply the root test: \( a_n = \left(\frac{(n^2+3n)^n}{(4n^2+5)^n}\right)^{1/n} = \frac{n^2+3n}{4n^2+5} \).For part b, the series is \( \sum_{n=1}^{\infty} b_n = \sum_{n=1}^{\infty} \frac{n^n}{(\ln(n))^n} \). Here, \( b_n = \left(\frac{n^n}{(\ln(n))^n} \right)^{1/n} = \frac{n}{\ln(n)} \).

Apply the Root Test for Part A

The root test involves computing \( L = \lim_{n \to \infty} \sqrt[n]{a_n} = \lim_{n \to \infty} \frac{n^2+3n}{4n^2+5} \). Simplify by dividing the numerator and denominator by \( n^2 \):\[L = \lim_{n \to \infty} \frac{1 + \frac{3}{n}}{4 + \frac{5}{n^2}}\]As \( n \to \infty \), \( \frac{3}{n} \to 0 \) and \( \frac{5}{n^2} \to 0 \). Hence, \( L = \frac{1}{4} \). Since \( L < 1 \), the series converges.

Apply the Root Test for Part B

Now compute \( L = \lim_{n \to \infty} \sqrt[n]{b_n} = \lim_{n \to \infty} \frac{n}{\ln(n)} \). For large \( n \), \( \ln(n) \to \infty \) but slower than \( n \), so the fraction \( \frac{n}{\ln(n)} \to \infty \).Thus, \( L = \infty \), which is greater than 1. Hence, the series diverges.

Key concepts

Root Test

The root test is a convergence test used in mathematical analysis to determine whether an infinite series converges or diverges. To apply the root test, we start by considering a series of the form \( \sum_{n=1}^{\infty} a_n \). We then compute the limit\[ L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \].
Here's what the results tell us:

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), or the limit goes to infinity, the series diverges.
• If \( L = 1 \), the root test is inconclusive, meaning we can't draw a conclusion using this test alone.

The root test is particularly useful for series with terms containing powers of \( n \). It can simplify the process by quickly showing whether terms are decreasing at a geometric rate, which leads to convergence.

Series Convergence

Series convergence is a fundamental concept in calculus and mathematical analysis. It is used to understand when the sum of an infinite series results in a finite number.
An infinite series \( \sum_{n=1}^{\infty} a_n \) converges if the sequence of partial sums \( S_n = \sum_{k=1}^{n} a_k \) approaches a limit as \( n \) becomes infinitely large. Simply put, convergence occurs when the sum gets closer and closer to a specific value, without exceeding it.
Different tests, like the root test, ratio test, and integral test, are used to evaluate convergence. Each test works best with certain types of series based on the nature of their terms.
Convergence is crucial because it tells us whether a series has a sum that can be meaningfully interpreted or used in calculations, especially in cases where series represent real-world phenomena or functions in mathematical models.

Mathematical Analysis

Mathematical analysis is a branch of mathematics that focuses on limits, continuity, and infinite series, among other topics. It provides the foundational tools needed to rigorously study the behavior of functions and sequences.
A key component of mathematical analysis involves examining the convergence of sequences and series, which can be thought of as infinite 'sums' built from sequences. This involves using various convergence tests, such as the root test, to determine if these series have a determinate sum.
Analysis goes beyond basic calculus through deeper exploration of concepts like real and complex number systems, differentiability, and integration. It also paves the way to more advanced topics like functional analysis and measure theory.
An understanding of mathematical analysis is crucial not only for pure mathematics but also in applied fields such as physics, engineering, and economics, where precise calculations and predictions are needed.