Evaluate The Series Or State That It Diverges

Evaluating whether aseries converges or diverges is a fundamental task in calculus and mathematical analysis. A series is essentially the sum of an infinite sequence of numbers. While some series sum to a finite value (converge), others grow without bound (diverge). Determining the behavior of a series is crucial across mathematics, physics, engineering, and computer science. This guide provides a structured approach to evaluating series convergence or divergence.

Introduction

A series (\sum_{n=1}^{\infty} a_n) is defined as the sum of the terms (a_1, a_2, a_3, \ldots). Convergence means the partial sums (S_n = a_1 + a_2 + \ldots + a_n) approach a finite limit as (n) approaches infinity. Divergence occurs when these partial sums either grow without bound (tending to (+\infty) or (-\infty)) or oscillate without settling to a single value. Evaluating a series involves systematically applying tests to determine its behavior. The most common tests include the Divergence Test, Ratio Test, Root Test, Comparison Test, Limit Comparison Test, Integral Test, Alternating Series Test, and the p-Series Test.

Steps for Evaluating Series Convergence or Divergence

1. Apply the Divergence Test (First Check):

   • Step: Calculate the limit of the general term (L = \lim_{n \to \infty} a_n).
   • Action: If (L) is not equal to zero (i.e., (L = c \neq 0) or (L = \pm \infty)), the series diverges. This is often the quickest test to apply.
   • Example: For (\sum \frac{n+1}{n}), (L = \lim_{n \to \infty} \frac{n+1}{n} = 1 \neq 0), so the series diverges.

2. Identify the Series Type (If Applicable):

   • Step: Look for recognizable patterns. Is it a geometric series? An alternating series? A p-series?
   • Action: If the series matches a known convergent or divergent form, you can often conclude directly.
   • Geometric Series: (\sum ar^n). Converges if (|r| < 1) (to (\frac{a}{1-r})), diverges if (|r| \geq 1).
   • p-Series: (\sum \frac{1}{n^p}). Converges if (p > 1), diverges if (p \leq 1).
   • Alternating Series: (\sum (-1)^n b_n) or (\sum (-1)^{n+1} b_n) with (b_n > 0). Converges if (b_n) is decreasing and (\lim_{n \to \infty} b_n = 0) (Alternating Series Test).
   • Example: (\sum \frac{1}{n^2}) is a p-series with (p=2>1), so it converges.

3. Apply the Ratio Test (For Factorials, Exponentials, Powers):

   • Step: Calculate (L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|).
   • Action: If (L < 1), the series converges absolutely. If (L > 1), the series diverges. If (L = 1), the test is inconclusive.
   • Example: For (\sum \frac{n!}{n^n}), (L = \lim_{n \to \infty} \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!} = \lim_{n \to \infty} \frac{n^n}{(n+1)^n} \cdot \frac{n+1}{n+1} = \lim_{n \to \infty} \left(\frac{n}{n+1}\right)^n \cdot 1 = \lim_{n \to \infty} \left(1 - \frac{1}{n+1}\right)^n = \frac{1}{e} < 1), so it converges.

4. Apply the Root Test (For Powers of n):

   • Step: Calculate (L = \lim_{n \to \infty} \sqrt[n]{|a_n|}).
   • Action: If (L < 1), the series converges absolutely. If (L > 1), the series diverges. If (L = 1), the test is inconclusive.
   • Example: For (\sum \left(\frac{n}{2n+1}\right)^n), (L = \lim_{n \to \infty} \sqrt[n]{\left(\frac{n}{2n+1}\right)^n} = \lim_{n \to \infty} \frac{n}{2n+1} = \frac{1}{2} < 1), so it converges.

5. Apply the Comparison Test (For Positive Series):

   • Step: Find a series (\sum b_n) with known convergence/divergence and terms (b_n > 0) that are comparable to (a_n).
   • Action: If (0 \leq a_n \leq b_n) for all (n) sufficiently large:
     • If (\sum b_n) converges, then (\sum a_n) converges.
     • If (\sum a_n \geq b_n) for all (n) sufficiently large and (\sum b_n) diverges, then (\sum a_n) diverges.
   • Example: For (\sum \frac{1}{n^2 + n}), compare to (\sum \frac{1}{n^2}). Since (\frac{1}{n^2 + n} < \frac{1}{n^2}) and (\sum \frac{1}{n^2}) converges, (\sum \frac{1}{n^2 + n}) converges.

6. Apply the Limit Comparison Test (For Positive Series):

   • Step: Calculate (L = \lim_{n \to \infty} \frac{a_n}{b_n}) where (b_n > 0) and (\sum b_n) converges or diverges.
   • Action: If (0 < L < \infty), then (\sum a_n) and (\sum b_n) both converge or both diverge.
   • Example: For (\sum \frac{n}{n^3 + 1}), compare to (\sum \frac{1}{n^2}). (L = \lim_{n \to \infty} \frac{n/(n^3+1)}{1/n^2} = \lim_{n \to \infty} \frac