Series

\begin{matrix} \sum_{κ = 1}^{6} 1 & = & \underset{\underset{6}{⏟}}{1 + 1 + 1 + 1 + 1 + 1} \\ \sum_{κ = 1}^{6} κ & = & 1 + 2 + \dots + 6 \\ \sum_{κ = 1}^{1000} 3^{κ} & = & 3^{1} + 3^{2} + \dots + 3^{1000} \end{matrix}

\begin{matrix} \sum_{κ = 1}^{\infty} κ & = & 1 + 2 + 3 + \dots + n + \dots \\ s_{1} & = & 1 = \sum_{κ = 1}^{1} κ \\ s_{2} & = & 3 = 1 + 2 = \sum_{κ = 1}^{2} κ \\ s_{3} & = & 6 = 1 + 2 + 3 = \sum_{κ = 1}^{3} κ \\ ⋮ \\ s_{n} & = & \frac{n (n + 1)}{2} = 1 + 2 + 3 + \dots + n = \sum_{κ = 1}^{n} κ \end{matrix}

Terminology

Infinite series

An infinite series is the sum of an infinite sequence

\sum_{κ = 1}^{\infty} 𝒶_{κ}

The value $𝒶_{κ}$ is the $κ$ 'th term of the series.

Partial sums

The sum $s_{n}$ of the first $n$ -terms of a series $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ is the $n^{th}$ partial sum. The sequence ${s_{n}}$

\begin{matrix} s_{1} & = & 𝒶_{1} \\ s_{2} & = & 𝒶_{1} + 𝒶_{2} = \sum_{κ = 1}^{2} 𝒶_{κ} \\ ⋮ \\ s_{n} & = & 𝒶_{1} + 𝒶_{2} + 𝒶_{3} + \dots + 𝒶_{n} = \sum_{κ = 1}^{n} 𝒶_{κ} \\ ⋮ \end{matrix}

is the sequence of partial sums.

Convergence

For a series $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ , if ${s_{n}} \to 𝒮$ then the series converges and

\sum_{κ = 1}^{\infty} 𝒶_{κ} = 𝒮

If the series is not convergent then it is the divergent.

Convergence properties

If $\sum_{κ = 1}^{\infty} 𝒶_{κ} = 𝒜$ , $\sum_{κ = 1}^{\infty} 𝒷_{κ} = ℬ$ and $α, β \in ℝ$ then

\sum_{κ = 1}^{\infty} (α 𝒶_{κ} + β 𝒷_{κ}) = α 𝒜 + β ℬ

Examples

Part I

Examples

\begin{matrix} \sum_{κ = 1}^{\infty} \frac{1}{2^{κ}} & \sum_{κ = 1}^{\infty} \frac{1}{κ (κ + 1)} \\ \sum_{κ = 1}^{\infty} \frac{κ}{κ + 1} & \sum_{κ = 1}^{\infty} {(-1)}^{κ} \end{matrix}

Geometric series

\sum_{κ = 1}^{\infty} α 𝓇^{κ}

Harmonic series

\sum_{κ = 1}^{\infty} \frac{1}{κ}

Properties

\sum_{κ = 1}^{\infty} (\frac{1}{2^{κ - 3}} - \frac{3}{κ (κ + 1)})

Integral test

\begin{matrix} \sum_{κ = 1}^{\infty} \frac{1}{κ^{2}} & \sum_{κ = 1}^{\infty} \frac{1}{\sqrt[3]{κ^{2}}} \\ \sum_{κ = 1}^{\infty} \frac{1}{\sqrt{2 κ - 1}} & \sum_{κ = 1}^{\infty} \frac{1}{\sqrt{κ^{3}}} \end{matrix}

Remainder estimate

Approximate $\sum_{κ = 1}^{\infty} \frac{1}{κ^{2}}$ to within an error bound of $0.005$ .

Divergence test

Integral tests

Necessary convergence condition

If $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ converges then ${𝒶_{n}} \to 0$ .

Divergence test

If ${𝒶_{n}} ↛ 0$ then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ is divergent.

Area comparison

Integral

Integral test

Let ${𝒶_{κ}}$ be a sequence with positive terms and $f (x)$ be continuous, positive and decreasing real valued function. Suppose also there is $N_{0} \in ℕ$ such that

\forall κ \in ℕ N_{0} \leq κ \Rightarrow f (κ) = 𝒶_{κ}

then

\sum_{κ = N_{0}}^{\infty} 𝒶_{κ} and \int_{N_{0}}^{\infty} f (x) ⅆ x

either both diverge or both converge.

$p$ -series test

The $p$ -series $\sum_{κ = 1}^{\infty} \frac{1}{κ^{p}}$ is convergent for $p > 1$ and divergent otherwise.

$n^{th}$ –remainder

The $n^{th}$ –remainder $ℛ_{n}$ of a series $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ is

ℛ_{n} = \sum_{κ = n + 1}^{\infty} 𝒶_{κ}

$n^{th}$ –remainder estimate

If $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ is convergent then

\int_{n + 1}^{\infty} f (x) ⅆ x \leq ℛ_{n} \leq \int_{n}^{\infty} f (x) ⅆ x

Comparison Tests

Area comparison

Comparison test

Let $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ and $\sum_{κ = 1}^{\infty} 𝒷_{κ}$ be series with non-negative terms. Suppose also there is $N_{0} \in ℕ$ such that

\forall κ \in ℕ N_{0} \leq κ \Rightarrow 𝒶_{κ} \leq 𝒷_{κ}

then

if $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ diverges, then $\sum_{κ = 1}^{\infty} 𝒷_{κ}$ diverges;
if $\sum_{κ = 1}^{\infty} 𝒷_{κ}$ converges, then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ converges.

Limit comparison test

Positive limit

Let $𝒶_{κ} > 0$ and $𝒷_{κ} > 0$ for all $κ \geq N_{0} \in ℕ$ , let

\lim_{κ \to \infty} \frac{𝒶_{κ}}{𝒷_{κ}} = ℒ

then

if $ℒ > 0$: then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ and $\sum_{κ = 1}^{\infty} 𝒷_{κ}$ both converge or both diverge;

Limit comparison test

Zero or infinite limit

if $ℒ = 0$ and $\sum_{κ = 1}^{\infty} 𝒷_{κ}$ converges: then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ converges;
if $ℒ = \infty$ and $\sum_{κ = 1}^{\infty} 𝒷_{κ}$ diverges: then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ diverges.

Examples

Part II

Comparison tests

\begin{matrix} \sum_{κ = 1}^{\infty} \frac{1}{2^{κ} + 3} & \sum_{κ = 1}^{\infty} \frac{1}{κ^{3} + 3 κ + 1} \\ \sum_{κ = 1}^{\infty} \frac{2}{ln κ} & \sum_{κ = 1}^{\infty} \frac{1}{κ^{2} - 1} \end{matrix}

Limit tests

\begin{matrix} \sum_{κ = 1}^{\infty} \frac{1}{κ^{2} - 1} & \sum_{κ = 1}^{\infty} \frac{2^{κ}}{3^{κ} - 1} \\ \sum_{κ = 2}^{\infty} \frac{ln κ}{κ^{2}} & \sum_{κ = 2}^{\infty} \frac{1}{\sqrt{κ} - 1} \end{matrix}

Alternating series

Definition

A series whose terms alternate between positive and negative is called alternating series, i.e.,

\begin{matrix} \sum_{κ = 1}^{\infty} {(-1)}^{κ + 1} 𝒷_{κ} & = & 𝒷_{1} - 𝒷_{2} + 𝒷_{3} - 𝒷_{4} + \dots + {(-1)}^{κ + 1} 𝒷_{κ} + \dots \\ or \\ \sum_{κ = 1}^{\infty} {(-1)}^{κ} 𝒷_{κ} & = & - 𝒷_{1} + 𝒷_{2} - 𝒷_{3} + 𝒷_{4} - \dots + {(-1)}^{κ} 𝒷_{κ} + \dots \end{matrix}

where $𝒷_{κ} > 0$ for all $κ$ .

Convergence

The series

\sum_{κ = 1}^{\infty} {(-1)}^{κ + 1} 𝒷_{κ}

converges if

$𝒷_{κ} > 0$
${𝒷_{κ}}$ is monotone decreasing i.e., $𝒷_{κ} \geq 𝒷_{κ + 1}$
${𝒷_{κ}} \to 0$

Remainder

If $𝒮 = \sum_{κ = 1}^{\infty} {(-1)}^{κ + 1} 𝒷_{κ}$ and

$𝒷_{κ} > 0$
${𝒷_{κ}}$ is monotone decreasing i.e., $𝒷_{κ} \geq 𝒷_{κ + 1}$
${𝒷_{κ}} \to 0$

then

| ℛ_{n} | = | 𝒮 - s_{n} | \leq 𝒷_{n}

Absolute convergence

A series

\sum_{κ = 1}^{\infty} 𝒶_{κ}

converges absolutely or is absolutely convergent if

\sum_{κ = 1}^{\infty} | 𝒶_{κ} |

converges.

Absolute implies conditional

If $\sum_{κ = 1}^{\infty} | 𝒶_{κ} |$ converges then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ converges.

Examples

Part III

Alternating series

\begin{matrix} \sum_{κ = 1}^{\infty} {(-1)}^{κ} & \sum_{κ = 1}^{\infty} \frac{{(-1)}^{κ}}{κ} \\ \sum_{κ = 1}^{\infty} \frac{{(-1)}^{κ}}{κ^{2} - 1} & \sum_{κ = 1}^{\infty} \frac{{(-1)}^{κ} κ}{κ + 1} \end{matrix}

Remainder estimate

Approximate $\sum_{κ = 1}^{\infty} \frac{{(-1)}^{κ}}{κ!}$ correct up to three decimal points.

Conditional vs absolute

\begin{matrix} \sum_{κ = 1}^{\infty} \frac{{(-1)}^{κ + 1}}{3 κ + 1} & \sum_{κ = 1}^{\infty} \frac{sin κ}{κ^{2}} \end{matrix}

Ratio and root tests

\begin{matrix} \sum_{κ = 1}^{\infty} \frac{1}{κ} & \sum_{κ = 1}^{\infty} \frac{1}{κ^{2}} \\ \sum_{κ = 1}^{\infty} \frac{2^{κ} + 5}{3^{κ}} & \sum_{κ = 1}^{\infty} \frac{{(-1)}^{κ} κ! κ!}{(2 κ)!} \\ \sum_{κ = 1}^{\infty} \frac{κ^{κ}}{κ!} & \sum_{κ = 1}^{\infty} \frac{1}{κ^{κ}} \end{matrix}

Ratio and root tests

\begin{matrix} \sum_{κ = 1}^{\infty} 𝒶_{κ} & where 𝒶_{κ} = {\begin{matrix} \frac{κ}{2^{κ}} & κ = 2 m + 1 \\ \frac{1}{2^{κ}} & κ = 2 m \end{matrix} \\ \sum_{κ = 1}^{\infty} \frac{2^{κ}}{κ^{3}} \end{matrix}

Ratio and Root test

Ratio test

Let $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ be a series with positive terms and let

\lim_{κ \to \infty} \frac{𝒶_{κ + 1}}{𝒶_{κ}} = ρ

then

if $ρ < 1$ then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ converges;
if $ρ > 1$ or $ρ = ∞$ then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ diverges;
if $ρ = 1$ test is inconclusive.

Root test

Let $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ be a series with positive terms and let

\lim_{κ \to \infty} \sqrt[κ]{𝒶_{κ}} = ρ

then

if $ρ < 1$ then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ converges;
if $ρ > 1$ or $ρ = ∞$ then $\sum_{κ = 1}^{\infty} 𝒶_{κ}$ diverges;
if $ρ = 1$ test is inconclusive.

Series

Terminology

Infinite series

Partial sums

Convergence

Convergence properties

Examples

Part I

Examples

Geometric series

Harmonic series

Properties

Integral test

Remainder estimate

Divergence test

Integral tests

Necessary convergence condition

Divergence test

Area comparison

Integral

Integral test

p -series test

nth–remainder

nth–remainder estimate

Comparison Tests

Area comparison

Comparison test

Limit comparison test

Positive limit

Limit comparison test

Zero or infinite limit

Examples

Part II

Comparison tests

Limit tests

Alternating series

Definition

Convergence

Remainder

Absolute convergence

Absolute implies conditional

Examples

Part III

Alternating series

Remainder estimate

Conditional vs absolute

Ratio and root tests

Ratio and root tests

Ratio and Root test

Ratio test

Root test

$p$ -series test

$n^{th}$ –remainder

$n^{th}$ –remainder estimate