Sequences and Series – Dr. Mohamed Mabrok

Sequences, Series & Convergence Tests

Understanding the fundamentals of infinite processes in mathematics

Introduction

Sequences

A sequence is an ordered list of numbers. Formally, a sequence is a function whose domain is the set of natural numbers.

a_n = \frac{1}{n} \quad \text{for } n = 1, 2, 3, \ldots

We say a sequence converges to a limit \( L \) if the terms \( a_n \) get arbitrarily close to \( L \) as \( n \) increases.

Series

A series is the sum of the terms of a sequence. An infinite series is the sum of infinitely many terms.

\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots

A series converges if the sequence of partial sums \( S_N = \sum_{n=1}^{N} a_n \) converges to a finite limit.

Convergence Tests Explained

Divergence Test

If the limit of the sequence terms doesn't approach zero, the series must diverge.

\text{If } \lim_{n \to \infty} a_n \neq 0, \text{ then } \sum a_n \text{ diverges.}

Example:

Does \( \sum_{n=1}^{\infty} \frac{n}{n+1} \) converge?

\lim_{n \to \infty} \frac{n}{n+1} = 1 \neq 0

The series diverges by the Divergence Test.

Geometric Series Test

A geometric series converges if the common ratio has absolute value less than 1.

\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \text{ if } |r| < 1

Example:

Does \( \sum_{n=0}^{\infty} \frac{2}{3^n} \) converge?

\text{This is geometric with } a = 2, r = \frac{1}{3}, |r| < 1 \] \[ \text{Sum} = \frac{2}{1-\frac{1}{3}} = 3

The series converges to 3.

p-Series Test

A p-series converges if the exponent p is greater than 1.

\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges if } p > 1

Example:

Does \( \sum_{n=1}^{\infty} \frac{1}{n^{1.5}} \) converge?

\[ p = 1.5 > 1 \]

The series converges by the p-Series Test.

Comparison Test

Compare to a known series. If your series is smaller than a convergent series, it converges. If larger than a divergent series, it diverges.

\text{If } 0 \leq a_n \leq b_n \text{ and } \sum b_n \text{ converges, then } \sum a_n \text{ converges.}

Example:

Does \( \sum_{n=1}^{\infty} \frac{1}{n^2 + 1} \) converge?

\frac{1}{n^2 + 1} \leq \frac{1}{n^2} \text{ and } \sum \frac{1}{n^2} \text{ converges (p=2)}

The series converges by Comparison Test.

Ratio Test

Examine the limit of the ratio of consecutive terms. Useful for factorials and exponentials.

\[ \text{Let } L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \] If L < 1: converges, L > 1: diverges, L = 1: inconclusive

Example:

Does \( \sum_{n=1}^{\infty} \frac{n!}{10^n} \) converge?

L = \lim_{n \to \infty} \frac{(n+1)!/10^{n+1}}{n!/10^n} = \lim_{n \to \infty} \frac{n+1}{10} = \infty

The series diverges by Ratio Test (L > 1).

Root Test

Examine the limit of the nth root of the absolute value of the terms. Useful when terms have nth powers.

\[ \text{Let } L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \] If L < 1: converges, L > 1: diverges, L = 1: inconclusive

Example:

Does \( \sum_{n=1}^{\infty} \left(\frac{2n}{3n+1}\right)^n \) converge?

L = \lim_{n \to \infty} \sqrt[n]{\left(\frac{2n}{3n+1}\right)^n} = \lim_{n \to \infty} \frac{2n}{3n+1} = \frac{2}{3} < 1

The series converges by Root Test.

Alternating Series Test

For series with alternating signs, if the absolute values decrease to zero, the series converges.

\sum (-1)^{n+1} a_n \text{ converges if } a_n \text{ decreases and } \lim_{n \to \infty} a_n = 0

Example:

Does \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \) converge?

\frac{1}{n} \text{ decreases and } \lim_{n \to \infty} \frac{1}{n} = 0

The series converges by Alternating Series Test.

Integral Test

If the function f(n) = aₙ is continuous, positive, and decreasing, the series behaves like its integral.

\sum_{n=1}^{\infty} a_n \text{ and } \int_{1}^{\infty} f(x) dx \text{ both converge or both diverge}

Example:

Does \( \sum_{n=1}^{\infty} \frac{1}{n \ln n} \) converge?

\int_{2}^{\infty} \frac{1}{x \ln x} dx = \lim_{b \to \infty} \ln(\ln x) \big|_{2}^{b} = \infty

The series diverges by Integral Test.

Summary Table

Test	When to Use	Conditions	Conclusion
Divergence Test	First test for any series	lim aₙ ≠ 0	Series diverges
Geometric Series	Terms are constant multiples of rⁿ	\|r\| < 1	Converges to a/(1-r)
p-Series	Terms are 1/nᵖ	p > 1	Converges
Comparison Test	Can compare to known series	0 ≤ aₙ ≤ bₙ	If ∑bₙ converges, then ∑aₙ converges
Ratio Test	Factorials or exponentials	lim \|aₙ₊₁/aₙ\| = L	L < 1: converges, L > 1: diverges
Root Test	Terms have nth powers	lim ⁿ√\|aₙ\| = L	L < 1: converges, L > 1: diverges
Alternating Series	Signs alternate	\|aₙ\| decreases to 0	Converges
Integral Test	f(n) = aₙ is integrable	f positive, continuous, decreasing	∫f and ∑aₙ same behavior