Convergence and Divergence Tests for Series Review

Convergence and Divergence Tests for Series Review

We will now review all of the convergence and divergence tests for series.

Let $\displaystyle{\sum_{n=1}^{\infty} a_n}$ be a series.

Convergence and Divergence Tests for Series

Test Summary
Sequence of Terms Divergence Criterion for Infinite Series If $\displaystyle{\lim_{n \to \infty} a_n \neq 0}$ then the series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.
Cauchy's Condition for Convergent Series The series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges if and only if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ and for all $p \in \mathbb{N}$ we have that $\mid a_{n+1} + a_{n+2} + ... + a_{n+p} \mid < \epsilon$.
The Alternating Series Test for Alternating Series of Real Numbers If $(a_n)_{n=1}^{\infty}$ is a sequence of positive real numbers where $(a_n)_{n=1}^{\infty}$ is a decreasing sequence and $\displaystyle{\lim_{n \to \infty} a_n = 0}$ then the alternating series $\displaystyle{\sum_{n=1}^{\infty} (-1)^n a_n}$ converges.
Absolute and Conditional Convergence of Series of Real Numbers If $\displaystyle{\sum_{n=1}^{\infty} \mid a_n \mid}$ converges then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.
Geometric Series of Real Numbers The geometric series $\displaystyle{\sum_{n=1}^{\infty} ax^{n-1}}$ converges to $\displaystyle{\frac{a}{1 - x}}$ if $\mid x \mid < 1$ and diverges if $\mid x \mid \geq 1$.
The Comparison Test for Positive Series of Real Numbers If $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are positive sequences of real numbers such that $a_n \leq b_n$ ultimately then:
a) If $\displaystyle{\sum_{n=1}^{\infty} b_n}$ converges then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.
b) If $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges then $\displaystyle{\sum_{n=1}^{\infty} b_n}$ diverges.
The Limit Comparison Test for Positive Series of Real Numbers If $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are positive sequences of real numbers and $\displaystyle{L = \lim_{n \to \infty} \frac{a_n}{b_n}}$ then:
a) If $0 < L < \infty$ then both series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ and $\displaystyle{\sum_{n=1}^{\infty} b_n}$ either both converge or both diverge.
b) If $L = 0$ and $\displaystyle{\sum_{n=1}^{\infty} b_n}$ converges then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.
c) If $L = \infty$ and $\displaystyle{\sum_{n=1}^{\infty} b_n}$ diverges then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.
The Limit Superior and Limit Inferior Comparison Test for Positive Series of Real Numbers If $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are positive sequences of real numbers, $\displaystyle{L^* = \limsup_{n \to \infty} \left \{ \frac{a_n}{b_n} \right \} }$, and $\displaystyle{L_*= \liminf_{n \to \infty} \left \{ \frac{a_n}{b_n} \right \} }$. Then:
a) If $L^* < \infty$ and $\displaystyle{\sum_{n=1}^{\infty} b_n}$ converges then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.
b) If $L_* > 0$ and $\displaystyle{\sum_{n=1}^{\infty} b_n}$ diverges then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.
The Integral Test for Positive Series of Real Numbers If $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is a positive series and there exists a positive integrable function $f$ that decreases and converges to $0$ on $[1, \infty)$ such that $f(n) = a_n$ for all $n \in \mathbb{N}$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges if and only if $\displaystyle{\int_1^{\infty} f(x) \: dx}$ converges.
The Ratio Test for Positive Series of Real Numbers If $(a_n)_{n=1}^{\infty}$ is a positive sequence of real numbers, $\displaystyle{\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho}$, then:
a) If $0 \leq \rho < 1$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.
b) If $1 < \rho \leq \infty$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.
c) If $\rho = 1$ then this test is inconclusive.
The Root Test for Positive Series of Real Numbers If $(a_n)_{n=1}^{\infty}$ is a positive sequence of real numbers and$\displaystyle{\lim_{n \to \infty} (a_n)^{1/n} = L}$ then:
a) If $0 \leq L < 1$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.
b) If $1 < L \leq \infty$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.
c) If $L = 1$ then this test is inconclusive.
The Limit Superior/Inferior Ratio Test for Series of Complex Numbers If $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is a series of nonzero complex numbers, $\displaystyle{R = \limsup_{n \to \infty} \biggr \lvert \frac{a_n}{b_n} \biggr \rvert}$, and $\displaystyle{r = \liminf_{n \to \infty} \biggr \lvert \frac{a_n}{b_n} \biggr \rvert}$. Then:
a) If $R < 1$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges.
b) If $1 < r$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.
c) If $r \leq 1 \leq R$ then this test is inconclusive.
The Limit Superior Root Test for Series of Complex Numbers If $\displaystyle{\sum_{n=1}^{\infty} a_n}$ be a series of complex numbers and let $\displaystyle{L = \limsup_{n \to \infty} \left ( \mid a_n \mid \right )^{1/n}}$. Then:
a) If $0 \leq L < 1$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges absolutely.
b) If $1 < L \leq \infty$ then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ diverges.
c) If $L = 1$ then this test is inconclusive.
Dirichlet's Test for Convergence of Series of Real Numbers If $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ are two sequences of real numbers, the sequence of partial sums of $(a_n)_{n=1}^{\infty}$, $(A_n)_{n=1}^{\infty}$, is bounded and $(b_n)_{n=1}^{\infty}$ is a decreasing sequence such that $\displaystyle{\lim_{n \to \infty} b_n = 0}$ then $\displaystyle{\sum_{n=1}^{\infty} a_nb_n}$ converges.
Abel's Test for Convergence of Series of Real Numbers If $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of real numbers, $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges, and $(b_n)_{n=1}^{\infty}$ is monotonically convergent then $\displaystyle{\sum_{n=1}^{\infty} a_nb_n}$ converges.