Convergence of real series

$Infinite series$

Infinite series of

{a_{n}}_{n = 1}^{\infty}

: \sum_{n = 1}^{\infty} a_{n}

n

-partial sum of

{a_{n}}

: S_{n} = \sum_{k = 1}^{n} a_{k}

\sum_{n = 1}^{\infty} a_{n}

converge :

{S_{n}}_{n = 1}^{\infty}

converges

\sum_{n = 1}^{\infty} a_{n}

diverge :

{S_{n}}_{n = 1}^{\infty}

diverges

1.∣ r ∣ < 1 \Rightarrow n = 1 \sum \infty a r^{n = 1} = \frac{a}{1 - r} 2.∣ r ∣ \geq 1 \Rightarrow n = 1 \sum \infty a r^{n - 1} diverges

Alternating series

: n = 1 \sum \infty (- 1)^{n - 1} a_{n}, (\forall n \in N (a_{n} \geq 0))

a_{n} > 0 \land n \to \infty lim a_{n} = 0 \Rightarrow n = 1 \sum \infty (- 1)^{n - 1} a_{n}

converge

\forall n \in N (0 \leq a_{n} \leq b_{n}) \land n = 1 \sum \infty b_{n}

converge

\Rightarrow n = 1 \sum \infty a_{n}

converge

Harmonic series

: n = 1 \sum \infty \frac{1}{n ^{α}} (α > 0)

n = 1 \sum \infty \frac{1}{n ^{α}} (α > 0)

converge

: α > 1

n = 1 \sum \infty a_{n}

converge absolutely

: n = 1 \sum \infty ∣ a_{n} ∣

converge

n = 1 \sum \infty a_{n}

converge conditionally

: n = 1 \sum \infty ∣ a_{n} ∣

diverge

\land n = 1 \sum \infty a_{n}

converge

n = 1 \sum \infty ∣ a_{n} ∣

converge

\Rightarrow n = 1 \sum \infty a_{n}

converge

n \to \infty lim ∣ \frac{a _{n + 1}}{a _{n}} ∣ < 1 \Rightarrow n = 1 \sum \infty a_{n} absolutely converge n \to \infty lim ∣ \frac{a _{n + 1}}{a _{n}} ∣ > 1 \Rightarrow n = 1 \sum \infty a_{n} divergent

n \to \infty lim sup ∣ n a_{n} ∣ < 1 \Rightarrow n = 1 \sum \infty a_{n}

converges

n \to \infty lim sup ∣ n a_{n} ∣ > 1 \Rightarrow n = 1 \sum \infty a_{n}

diverges