Convergence Tests for Series

Topics: Calculus - Sequence - Series

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The convergence of series can be tested with the following tests:

Sequence Converging to 0 §

(theorem, condición del resto)

Let ${\sum }_{n=1}^{\infty }{a}_n$ be a convergent sequence, then:

$$\underset{x\to \infty }{\lim }{a}_n=0$$

This tells us that if the sequence that corresponds to a series doesn’t converge to $0$, then the series diverges.

Comparison Test §

(theorem)

Suppose that $0<a_n<b_n$ $\forall n\in \mathbb{N}$, and that ${\sum }_{n=1}^{\infty }{b}_n$ converges.

Then, ${\sum }_{n=1}^{\infty }{a}_n$ converges, and ${\sum }_{n=1}^{\infty }{a}_n<{\sum }_{n=1}^{\infty }{b}_n$.

Comparison Test 2 §

(theorem)

Let ${\sum }_{n=1}^{\infty }{a}_k$ and ${\sum }_{n=1}^{\infty }{b}_k$ be series, and $a_n,b_n>0$ $\forall n\in \mathbb{N}$. Furthermore, ${\lim }_{n\to \infty }\frac{a_n}{b_n}=c\ne 0$.

Then ${\sum }_{n=1}^{\infty }{a}_n$ converges if and only if ${\sum }_{n=1}^{\infty }{b}_n$ converges.

Ratio Test §

(theorem)

Let $a_n>0$ $\forall n\in \mathbb{N}$ and ${\lim }_{n\to \infty }\frac{a_{n+1}}{a_n}=r$.

Then, ${\sum }_{n=1}^{\infty }{a}_n$ converges if $r<1$ and diverges if $r>1$.

Integral Test §

(theorem)

Let $\{a_n{\}}_{n\in \mathbb{N}}$ be a sequence and $f:\mathbb{R}\to \mathbb{R}$ a non-negative decreasing function, such that $f(n)=a_n$ $\forall n\in \mathbb{N}$.

Then, ${\int }_1^{\infty }fdx$ (an improper integral) converges if and only if ${\sum }_{n=1}^{\infty }{a}_n$ converges.