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Definition/Summary
In what follows, we will work in a normed space [itex](X,\|~\|)[/itex].
A series is, by definition, two sequences [itex](u_n)_n[/itex] and [itex](s_n)_n[/itex] such that [itex]s_n=\sum_{k=0}^n{u_k}[/itex] for every n.
We call the elements [itex]u_n[/itex] the terms of the series. The elements [itex]s_n[/itex] are called the partial sums. We will often denote a series by [itex]\sum_{n=0}^{+\infty}{u_n}[/itex].
We say that a series [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges to a if and only if [itex]s_n\rightarrow a[/itex]. If a series does not converge, then we say that the series diverges.
Equations
Extended explanation
Series in a normed space
For the following, we will work in a normed space [itex](X,\| ~\|)[/itex]
nth term test
If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] is a series such that [tex]\lim_{n\rightarrow +\infty}{u_n}\neq 0,[/tex]
then the series diverges.
WARNING: The converse does not hold, i.e. if the limit does equal zero, then the series does not necessarily converge.
HINT: When given a series, always apply this test first.
Linearity of convergence
Let [itex]\lambda, \mu\in \mathbb{R}[/itex]. If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges to u and if [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges to v, then the series [itex]\sum_{n=0}^{+\infty}{\lambda u_n+\mu v_n}[/itex] converges to [itex]\lambda u+\mu v[/itex].
Deletion of finitely many terms
Let [itex]p\in \mathbb{N}[/itex]. Then we have the following equivalence:
[tex]\sum_{n=0}^{+\infty}{u_n}~\text{converges iff }~\sum_{n=p}^{+\infty}{u_n}~\text{converges}[/tex]
Series in a complete normed space
In the following, we will work in a Banach space (= a complete normed space).
Cauchy criterion
A series [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges if and only if
[tex]\forall \epsilon>0:~\exists n_0:~\forall n>n_0:~\forall p:~\left\|\sum_{k=n}^{n+p}{u_k}\right\|<\epsilon[/tex]
Absolute convergence
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] be a series. If the series [itex]\sum_{n=0}^{+\infty}{\|u_n\|}[/itex] converges, then the orginal series will converge. Moreover, we have
[tex]\left\|\sum_{n=0}^{+\infty}{u_n}\right\|\leq \sum_{n=0}^{+\infty}{\|u_n\|}[/tex]
NOTATION: A series such as in the above theorem is called absolutely convergent. Absolute convergence is handy because it allows you to transform a series to a series with positive real numbers.
Series with nonnegative real terms
In the following we will always work with series [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] such that all the [itex]u_n[/itex] are real and nonnegative.
Subseries
If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] is a convergent series and if [itex]\sum_{n=0}^{+\infty}{u_{k_n}}[/itex] is a subseries, then this subseries converges. In particular, we have that
[tex]\sum_{n=0}^{+\infty}{u_{k_n}}\leq \sum_{n=0}^{+\infty}{u_n}[/tex]
Comparison test
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] and [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] be two series such that [itex]u_n\leq v_n[/itex] for all n greater then a certain [itex]n_0[/itex]. Then we have:
1) If [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges, then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] diverges, then [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] diverges.
Limit comparison test
1) If [itex]\limsup_{n\rightarrow +\infty}{\frac{u_n}{v_n}}<+\infty[/itex] and if [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges, then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
2) If [itex]\liminf_{n\rightarrow +\infty}{\frac{u_n}{v_n}}>0[/itex] and if [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges, then [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges.
HINT: the limsup and liminf can be replaced by ordinary limits.
Comparison test 2
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] and [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] be series. If there exists an m such that for every [itex]n\geq m[/itex] it holds that [itex]\frac{u_{n+1}}{u_n}\leq \frac{v_{n+1}}{v_n}[/itex], then
1) If [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges, then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
1) If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] diverges, then [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] diverges.
Cauchy condensation test
Let [itex](u_n)_n[/itex] be a nonincreasing sequence, then
[tex]\sum_{n=0}^{+\infty}{u_n}~\text{converges if and only if}~\sum_{n=0}^{+\infty}{2^nu_{2^n}}~\text{converges.}[/tex]
Cauchy's root test
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] be a series. Then
1) If [itex]\limsup_{n\rightarrow +\infty}{\sqrt[n]{u_n}}<1[/itex], then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
2) If [itex]\limsup_{n\rightarrow +\infty}{\sqrt[n]{u_n}}>1[/itex], then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] diverges.
HINT: the limsup can be replaced by ordinary limits.
WARNING: if the limsup equals 1, then the test is inconclusive.
The ratio test of d'Alembert
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] be a series. Then
1) If [itex]\limsup_{n\rightarrow +\infty}{\frac{u_{n+1}}{u_n}}<1[/itex], then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
2) If [itex]\liminf_{n\rightarrow +\infty}{\frac{u_{n+1}}{u_n}}>1[/itex], then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] diverges.
HINT: the limsup and liminf can be replaced by ordinary limits.
WARNING: if the limits equal 1, then the test is inconclusive.
The integral test
Let [itex]f:[0,+\infty[\rightarrow\mathbb{R}^+[/itex] be a nonincreasing function. Then
[tex]\sum_{n=0}^{+\infty}{f(n)}~\text{converges if and only if}~\int_1^{+\infty}{f(x)dx}<+\infty[/tex]
ADDENDUM: If [itex]f:[0,+\infty[\rightarrow\mathbb{R}^+[/itex] is a nonincreasing function, then for every [itex]n\in \mathbb{N}[/itex] holds
[tex]\sum_{k=1}^n{f(k)}\leq \int_0^n{f(x)dx}\leq \sum_{k=0}^{n-1}{f(k)}[/tex]
Series in [itex]\mathbb{R}[/itex] and [itex]\mathbb{C}[/itex]
The criterion of Dirichlet
Let [itex]\sum_{n=0}^{+\infty}{a_n}[/itex] be a (real or complex) series such that it's sequence of partial sums is bounded. Let [itex](v_n)_n[/itex] be a nonincreasing sequence of real numbers which converges to 0. Then the sequence [itex]\sum_{n=0}^{+\infty}{v_na_n}[/itex] converges.
The criterion of Abel
Let [itex]\sum_{n=0}^{+\infty}{a_n}[/itex] be a (real or complex) convergent series . Let [itex](v_n)_n[/itex] be a bounded sequence of real numbers which is either nondecreasing or nonincreasing. Then the sequence [itex]\sum_{n=0}^{+\infty}{v_na_n}[/itex] converges.
The criterion of Leibniz
Let [itex](u_n)_n[/itex] be a nonincreasing sequence of real numbers which converges to 0. Then the series [itex]\sum_{n=0}^{+\infty}{(-1)^nu_n}[/itex] converges.
ADDENDUM: Denote [itex](s_n)_n[/itex] the partial sums of the series [itex]\sum_{n=0}^{+\infty}{(-1)^nu_n}[/itex] and denote s the limit of the series. Then the sequence [itex](s_{2n})_n[/itex] is nonincreasing and [itex](s_{2n+1})_n[/itex] is nondecreasing. Moreover, we have that [itex]|s-s_n|\leq u_n[/itex].
Some special series
Geometric series
Let x be an arbitrary real or complex number. Then
[tex]\sum_{n=0}^{+\infty}{x^n}~\text{converges if and only if}~|x|<1.[/tex]
Moreover, if the series converges, then [itex]\sum_{n=0}^{+\infty}{x^n}=\frac{1}{1-x}[/itex]
p-series
Let p be a real number. Then
[tex]\sum_{n=0}^{+\infty}{\frac{1}{n^p}}~\text{converges if and only if}~p>1.[/tex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
In what follows, we will work in a normed space [itex](X,\|~\|)[/itex].
A series is, by definition, two sequences [itex](u_n)_n[/itex] and [itex](s_n)_n[/itex] such that [itex]s_n=\sum_{k=0}^n{u_k}[/itex] for every n.
We call the elements [itex]u_n[/itex] the terms of the series. The elements [itex]s_n[/itex] are called the partial sums. We will often denote a series by [itex]\sum_{n=0}^{+\infty}{u_n}[/itex].
We say that a series [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges to a if and only if [itex]s_n\rightarrow a[/itex]. If a series does not converge, then we say that the series diverges.
Equations
Extended explanation
Series in a normed space
For the following, we will work in a normed space [itex](X,\| ~\|)[/itex]
nth term test
If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] is a series such that [tex]\lim_{n\rightarrow +\infty}{u_n}\neq 0,[/tex]
then the series diverges.
WARNING: The converse does not hold, i.e. if the limit does equal zero, then the series does not necessarily converge.
HINT: When given a series, always apply this test first.
Linearity of convergence
Let [itex]\lambda, \mu\in \mathbb{R}[/itex]. If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges to u and if [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges to v, then the series [itex]\sum_{n=0}^{+\infty}{\lambda u_n+\mu v_n}[/itex] converges to [itex]\lambda u+\mu v[/itex].
Deletion of finitely many terms
Let [itex]p\in \mathbb{N}[/itex]. Then we have the following equivalence:
[tex]\sum_{n=0}^{+\infty}{u_n}~\text{converges iff }~\sum_{n=p}^{+\infty}{u_n}~\text{converges}[/tex]
Series in a complete normed space
In the following, we will work in a Banach space (= a complete normed space).
Cauchy criterion
A series [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges if and only if
[tex]\forall \epsilon>0:~\exists n_0:~\forall n>n_0:~\forall p:~\left\|\sum_{k=n}^{n+p}{u_k}\right\|<\epsilon[/tex]
Absolute convergence
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] be a series. If the series [itex]\sum_{n=0}^{+\infty}{\|u_n\|}[/itex] converges, then the orginal series will converge. Moreover, we have
[tex]\left\|\sum_{n=0}^{+\infty}{u_n}\right\|\leq \sum_{n=0}^{+\infty}{\|u_n\|}[/tex]
NOTATION: A series such as in the above theorem is called absolutely convergent. Absolute convergence is handy because it allows you to transform a series to a series with positive real numbers.
Series with nonnegative real terms
In the following we will always work with series [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] such that all the [itex]u_n[/itex] are real and nonnegative.
Subseries
If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] is a convergent series and if [itex]\sum_{n=0}^{+\infty}{u_{k_n}}[/itex] is a subseries, then this subseries converges. In particular, we have that
[tex]\sum_{n=0}^{+\infty}{u_{k_n}}\leq \sum_{n=0}^{+\infty}{u_n}[/tex]
Comparison test
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] and [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] be two series such that [itex]u_n\leq v_n[/itex] for all n greater then a certain [itex]n_0[/itex]. Then we have:
1) If [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges, then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] diverges, then [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] diverges.
Limit comparison test
1) If [itex]\limsup_{n\rightarrow +\infty}{\frac{u_n}{v_n}}<+\infty[/itex] and if [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges, then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
2) If [itex]\liminf_{n\rightarrow +\infty}{\frac{u_n}{v_n}}>0[/itex] and if [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges, then [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges.
HINT: the limsup and liminf can be replaced by ordinary limits.
Comparison test 2
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] and [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] be series. If there exists an m such that for every [itex]n\geq m[/itex] it holds that [itex]\frac{u_{n+1}}{u_n}\leq \frac{v_{n+1}}{v_n}[/itex], then
1) If [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] converges, then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
1) If [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] diverges, then [itex]\sum_{n=0}^{+\infty}{v_n}[/itex] diverges.
Cauchy condensation test
Let [itex](u_n)_n[/itex] be a nonincreasing sequence, then
[tex]\sum_{n=0}^{+\infty}{u_n}~\text{converges if and only if}~\sum_{n=0}^{+\infty}{2^nu_{2^n}}~\text{converges.}[/tex]
Cauchy's root test
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] be a series. Then
1) If [itex]\limsup_{n\rightarrow +\infty}{\sqrt[n]{u_n}}<1[/itex], then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
2) If [itex]\limsup_{n\rightarrow +\infty}{\sqrt[n]{u_n}}>1[/itex], then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] diverges.
HINT: the limsup can be replaced by ordinary limits.
WARNING: if the limsup equals 1, then the test is inconclusive.
The ratio test of d'Alembert
Let [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] be a series. Then
1) If [itex]\limsup_{n\rightarrow +\infty}{\frac{u_{n+1}}{u_n}}<1[/itex], then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] converges.
2) If [itex]\liminf_{n\rightarrow +\infty}{\frac{u_{n+1}}{u_n}}>1[/itex], then [itex]\sum_{n=0}^{+\infty}{u_n}[/itex] diverges.
HINT: the limsup and liminf can be replaced by ordinary limits.
WARNING: if the limits equal 1, then the test is inconclusive.
The integral test
Let [itex]f:[0,+\infty[\rightarrow\mathbb{R}^+[/itex] be a nonincreasing function. Then
[tex]\sum_{n=0}^{+\infty}{f(n)}~\text{converges if and only if}~\int_1^{+\infty}{f(x)dx}<+\infty[/tex]
ADDENDUM: If [itex]f:[0,+\infty[\rightarrow\mathbb{R}^+[/itex] is a nonincreasing function, then for every [itex]n\in \mathbb{N}[/itex] holds
[tex]\sum_{k=1}^n{f(k)}\leq \int_0^n{f(x)dx}\leq \sum_{k=0}^{n-1}{f(k)}[/tex]
Series in [itex]\mathbb{R}[/itex] and [itex]\mathbb{C}[/itex]
The criterion of Dirichlet
Let [itex]\sum_{n=0}^{+\infty}{a_n}[/itex] be a (real or complex) series such that it's sequence of partial sums is bounded. Let [itex](v_n)_n[/itex] be a nonincreasing sequence of real numbers which converges to 0. Then the sequence [itex]\sum_{n=0}^{+\infty}{v_na_n}[/itex] converges.
The criterion of Abel
Let [itex]\sum_{n=0}^{+\infty}{a_n}[/itex] be a (real or complex) convergent series . Let [itex](v_n)_n[/itex] be a bounded sequence of real numbers which is either nondecreasing or nonincreasing. Then the sequence [itex]\sum_{n=0}^{+\infty}{v_na_n}[/itex] converges.
The criterion of Leibniz
Let [itex](u_n)_n[/itex] be a nonincreasing sequence of real numbers which converges to 0. Then the series [itex]\sum_{n=0}^{+\infty}{(-1)^nu_n}[/itex] converges.
ADDENDUM: Denote [itex](s_n)_n[/itex] the partial sums of the series [itex]\sum_{n=0}^{+\infty}{(-1)^nu_n}[/itex] and denote s the limit of the series. Then the sequence [itex](s_{2n})_n[/itex] is nonincreasing and [itex](s_{2n+1})_n[/itex] is nondecreasing. Moreover, we have that [itex]|s-s_n|\leq u_n[/itex].
Some special series
Geometric series
Let x be an arbitrary real or complex number. Then
[tex]\sum_{n=0}^{+\infty}{x^n}~\text{converges if and only if}~|x|<1.[/tex]
Moreover, if the series converges, then [itex]\sum_{n=0}^{+\infty}{x^n}=\frac{1}{1-x}[/itex]
p-series
Let p be a real number. Then
[tex]\sum_{n=0}^{+\infty}{\frac{1}{n^p}}~\text{converges if and only if}~p>1.[/tex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
