Determine Whether The Following Series Converges. Justify Your Answer

Determining Whether a Series Converges: A Practical Guide

Convergence is the heart of infinite series analysis. Knowing whether a series converges—or diverges—determines if the infinite sum has a finite value that can be used in calculus, physics, engineering, and beyond. This guide walks you through the key tests and concepts needed to decide convergence for any given series, complete with examples and justifications And it works..

Introduction

An infinite series is an expression of the form

[ \sum_{n=1}^{\infty} a_n ]

where (a_n) is the nth term. The series converges if the sequence of partial sums

[ S_N = \sum_{n=1}^{N} a_n ]

approaches a finite limit as (N \to \infty). Otherwise, the series diverges. Which means while some series are obvious (e. g.Plus, , (\sum 1/n^2) converges), many require deeper analysis. Below we outline a systematic approach: start with simple tests, move to more sophisticated ones, and understand the underlying theory.

1. Preliminary Checks

1.1 The Term Test (Test for Divergence)

• Rule: If (\lim_{n\to\infty} a_n \neq 0), the series diverges.
• Why it works: For a sum to settle at a finite value, the individual terms must eventually become negligible.
• Example: (\sum_{n=1}^{\infty} \frac{1}{n}). Since (\lim_{n\to\infty} 1/n = 0), the test is inconclusive; we need further analysis.

1.2 Absolute Value Check

If (\sum |a_n|) diverges, the original series may still converge conditionally (e.g., alternating harmonic series). If (\sum |a_n|) converges, the series converges absolutely, guaranteeing convergence.

2. Comparison Tests

2.1 Direct Comparison Test

Given a known benchmark series (\sum b_n) with (b_n \ge 0):

• If (0 \le a_n \le b_n) for all large (n) and (\sum b_n) converges, then (\sum a_n) converges.
• If (a_n \ge b_n \ge 0) for all large (n) and (\sum b_n) diverges, then (\sum a_n) diverges.

Example: (\sum \frac{1}{n^2 + 5}). Compare to (\sum \frac{1}{n^2}), which converges ((p=2>1)). Since (\frac{1}{n^2 + 5} \le \frac{1}{n^2}) for all (n), the series converges.

2.2 Limit Comparison Test

If (a_n, b_n > 0) and

[ \lim_{n\to\infty}\frac{a_n}{b_n} = L \quad (0 < L < \infty), ]

then (\sum a_n) and (\sum b_n) either both converge or both diverge.

Example: (\sum \frac{n^2}{n^5+1}). Compare to (\sum \frac{1}{n^3}). Compute the limit:

[ \lim_{n\to\infty}\frac{\frac{n^2}{n^5+1}}{\frac{1}{n^3}} = \lim_{n\to\infty}\frac{n^5}{n^5+1} = 1. ]

Since (\sum 1/n^3) converges ((p=3>1)), our series converges.

3. Integral Test

If (f(n)=a_n) is positive, continuous, and decreasing for (n \ge N), then

[ \sum_{n=N}^{\infty} a_n ]

converges iff

[ \int_{N}^{\infty} f(x),dx ]

converges.

Example: (\sum \frac{1}{n \ln n}) for (n \ge 2). Evaluate

[ \int_{2}^{\infty} \frac{1}{x \ln x},dx. ]

Let (u = \ln x); (du = \frac{1}{x}dx). That said, the integral becomes (\int_{\ln 2}^{\infty} \frac{1}{u},du = \infty). Hence, the series diverges Less friction, more output..

4. Ratio and Root Tests

4.1 Ratio Test

Compute

[ L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|. ]

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

Example: (\sum \frac{n!}{n^n}).

[ \frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(n+1)^{n+1}}\cdot\frac{n^n}{n!} = \frac{n^n}{(n+1)^n}\cdot\frac{1}{n+1}. ]

Simplify:

[ \left(\frac{n}{n+1}\right)^n \cdot \frac{1}{n+1} \to e^{-1}\cdot 0 = 0 < 1. ]

Thus the series converges.

4.2 Root Test

Compute

[ L = \lim_{n\to\infty}\sqrt[n]{|a_n|}. ]

Same conclusion rules as the ratio test. Useful when terms involve (n)-th powers And that's really what it comes down to. That's the whole idea..

5. Alternating Series Test (Leibniz)

For series (\sum (-1)^{n-1} b_n) where (b_n > 0):

• If (b_n) is decreasing and (\lim_{n\to\infty} b_n = 0), the series converges.
• Absolute convergence requires (\sum b_n) to converge.

Example: The alternating harmonic series (\sum (-1)^{n-1}\frac{1}{n}) converges, though (\sum 1/n) diverges.

6. Power Series and Radius of Convergence

For a power series

[ \sum_{n=0}^{\infty} c_n (x-a)^n, ]

the ratio test yields the radius (R):

[ \frac{1}{R} = \limsup_{n\to\infty}\sqrt[n]{|c_n|}. ]

• If (|x-a| < R), the series converges absolutely.
• If (|x-a| > R), the series diverges.
• Endpoint behavior must be tested separately.

Example: (\sum \frac{x^n}{n!}). Here (\sqrt[n]{|c_n|} = \sqrt[n]{1/n!} \to 0), so (R = \infty); the series converges for all real (x).

7. Practical Step‑by‑Step Decision Flow

1. Check the Term Test: If (\lim a_n \neq 0), stop—diverges.
2. Absolute Convergence?
   • Try the Ratio or Root test.
   • If inconclusive, use Comparison or Integral tests.
3. Conditional Convergence?
   • If terms alternate, apply Alternating Series Test.
   • For non‑alternating series, test absolute convergence first; if fails, the series diverges.
4. Special Cases:
   • p‑Series (\sum 1/n^p): converges if (p>1).
   • Geometric Series (\sum ar^n): converges if (|r|<1).
   • Harmonic Series (\sum 1/n): diverges.
5. Power Series: Compute radius (R); test endpoints separately.

8. Illustrative Examples

Example 1: (\displaystyle \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2})

1. Term test: (\lim 1/(n(\ln n)^2)=0).
2. Use the Integral Test:

[ \int_{2}^{\infty} \frac{1}{x(\ln x)^2},dx = \left[-\frac{1}{\ln x}\right]_{2}^{\infty} = \frac{1}{\ln 2} < \infty. ]

Thus the series converges.

Example 2: (\displaystyle \sum_{n=1}^{\infty} \frac{n!}{(2n)!})

Apply the Ratio Test:

[ \frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(2n+2)!}\cdot\frac{(2n)!}{n!} = \frac{(n+1)}{(2n+1)(2n+2)} \to 0. ]

Since (L=0<1), the series converges It's one of those things that adds up..

Example 3: (\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}})

1. Term test: (\lim 1/\sqrt{n} = 0).
2. Alternating Series Test: (b_n = 1/\sqrt{n}) is decreasing and tends to 0. Hence the series converges conditionally (because (\sum 1/\sqrt{n}) diverges).

9. Frequently Asked Questions (FAQ)

10. Conclusion

Determining the convergence of a series is a layered process that blends intuition with rigorous tests. Day to day, start with the simplest checks—term test, absolute convergence—and progress to comparison, integral, ratio, root, or alternating series tests as needed. Which means for power series, compute the radius of convergence and examine endpoints. Mastering these techniques equips you to tackle almost any infinite series confidently and accurately And it works..

11. Applications in Real-World Scenarios

The convergence of series extends far beyond theoretical mathematics, playing a critical role in fields such as physics, engineering, and computer science. Take this case: Fourier series—which decompose periodic functions into sums of sines and cosines—rely on convergence theorems