Variable rows. Absolute and conditional convergence Alternating series examples of solutions
A number series containing an infinite number of positive and an infinite number of negative terms is called alternating.
Absolute and conditional convergence
A series is called absolutely convergent if the series also converges.
If a series converges absolutely, then it is convergent (in the usual sense). The converse is not true.
A series is said to be conditionally convergent if it itself converges and the series composed of the modules of its members diverges.
Investigate for convergence series 
.
Let us apply the Leibniz sufficient test for alternating series. We get 
because the . Therefore, this series converges.
38. Alternating rows. Leibniz sign.
A special case of an alternating series is an alternating series, that is, a series in which successive terms have opposite signs.
Leibniz sign
For those alternating nearby, the Leibniz sufficient convergence test applies.
Let (an) be a number sequence such that
1. an+1< an для всех n;
Then alternating series are outgoing.
39. Functional rows. Power series. convergence radius. Convergence interval.
The concept of functional series and power series
The usual number series, remember, consists of numbers:
All members of the series are NUMBERS.
The functional row consists of FUNCTIONS:
In addition to polynomials, factorials and other gifts, the common term of the series certainly includes the letter "x". It looks like this, for example:
Like a number series, any functional series can be written in expanded form:
As you can see, all members of the functional series are functions.
The most popular type of functional series is power series.
Definition:
A power series is a series whose common term includes positive integer powers of the independent variable.
A simplified power series in many textbooks is written as follows: , where is the old familiar “stuffing” of number series (polynomials, degrees, factorials that depend only on “en”). The simplest example:
Let's look at this decomposition and rethink the definition: the members of the power series contain "x" in positive integer (natural) powers.
Very often, a power series can be found in the following "modifications": or where a is a constant. For example:
Strictly speaking, the simplified representations of the power series, or not quite correct. In the exponent, instead of the single letter "en", a more complex expression can be located, for example: 
Or this power series:
If only the exponents at "xAx" were natural.
Power Series Convergence.
Convergence interval, convergence radius and convergence area
There is no need to be afraid of such an abundance of terms, they go “next to each other” and are not particularly difficult to understand. It is better to choose some simple experimental series and immediately begin to understand.
I ask you to love and favor the power series The variable can take any real value from "minus infinity" to "plus infinity". Substitute several arbitrary x values into the common term of the series:
If x=1 then
If x=-1, then
If x=3 then
If x=-0.2, then
It is obvious that by substituting "x" into one or another value, we get different numerical series. Some number series will converge and some will diverge. And our task is to find the set of values \u200b\u200bof "x" at which the power series will converge. Such a set is called the region of convergence of the series.
For any power series (temporarily digressing from a specific example), three cases are possible:
1) The power series converges absolutely on some interval . In other words, if we choose any value of "x" from the interval and substitute it into the common term of the power series, then we get an absolutely convergent number series. Such an interval is called the interval of convergence of the power series.
The convergence radius, quite simply, is half the length of the convergence interval:
Geometrically, the situation looks like this: 
In this case, the interval of convergence of the series: the radius of convergence of the series: 
A number series containing an infinite number of positive and an infinite number of negative terms is called alternating.
Absolute and conditional convergence
A series is called absolutely convergent if the series also converges.
If a series converges absolutely, then it is convergent (in the usual sense). The converse is not true.
A series is said to be conditionally convergent if it itself converges and the series composed of the modules of its members diverges.
Investigate for convergence series 
.
Let us apply the Leibniz sufficient test for alternating series. We get 
because the . Therefore, this series converges.
38. Alternating rows. Leibniz sign.
A special case of an alternating series is an alternating series, that is, a series in which successive terms have opposite signs.
Leibniz sign
For those alternating nearby, the Leibniz sufficient convergence test applies.
Let (an) be a number sequence such that
1. an+1< an для всех n;
Then alternating series are outgoing.
39. Functional rows. Power series. convergence radius. Convergence interval.
The concept of functional series and power series
The usual number series, remember, consists of numbers:
All members of the series are NUMBERS.
The functional row consists of FUNCTIONS:
In addition to polynomials, factorials and other gifts, the common term of the series certainly includes the letter "x". It looks like this, for example:
Like a number series, any functional series can be written in expanded form:
As you can see, all members of the functional series are functions.
The most popular type of functional series is power series.
Definition:
A power series is a series whose common term includes positive integer powers of the independent variable.
A simplified power series in many textbooks is written as follows: , where is the old familiar “stuffing” of number series (polynomials, degrees, factorials that depend only on “en”). The simplest example:
Let's look at this decomposition and rethink the definition: the members of the power series contain "x" in positive integer (natural) powers.
Very often, a power series can be found in the following "modifications": or where a is a constant. For example:
Strictly speaking, the simplified representations of the power series, or not quite correct. In the exponent, instead of the single letter "en", a more complex expression can be located, for example: 
Or this power series:
If only the exponents at "xAx" were natural.
Power Series Convergence.
Convergence interval, convergence radius and convergence area
There is no need to be afraid of such an abundance of terms, they go “next to each other” and are not particularly difficult to understand. It is better to choose some simple experimental series and immediately begin to understand.
I ask you to love and favor the power series The variable can take any real value from "minus infinity" to "plus infinity". Substitute several arbitrary x values into the common term of the series:
If x=1 then
If x=-1, then
Definition 1
The number series $\sum \limits _(n=1)^(\infty )u_(n) $, whose members have arbitrary signs (+), (?), is called an alternating series.
The alternating series considered above are a special case of the alternating series; it is clear that not every alternating series is alternating. For example, the series $1-\frac(1)(2) -\frac(1)(3) +\frac(1)(4) +\frac(1)(5) -\frac(1)(6) - \frac(1)(7) +\ldots - $ alternating but not character-alternating series.
Note that in an alternating series of terms, both with the sign (+) and with the sign (-), there are infinitely many. If this is not true, for example, the series contains a finite number of negative terms, then they can be discarded and a series composed of only positive terms can be considered, and vice versa.
Definition 2
If the number series $\sum \limits _(n=1)^(\infty )u_(n) $ converges and its sum is S, and partial the sum is equal to $S_n$ , then $r_(n) =S-S_(n) $ is called the remainder of the series, and $\mathop(\lim )\limits_(n\to \infty ) r_(n) =\mathop(\ lim )\limits_(n\to \infty ) (S-S_(n))=S-S=0$, i.e. the remainder of the convergent series tends to 0.
Definition 3
A series $\sum \limits _(n=1)^(\infty )u_(n) $ is called absolutely convergent if the series composed of the absolute values of its members $\sum \limits _(n=1)^(\ infty )\left|u_(n) \right| $.
Definition 4
If the number series $\sum \limits _(n=1)^(\infty )u_(n) $ converges and the series $\sum \limits _(n=1)^(\infty )\left|u_(n )\right| $, composed of the absolute values of its members, diverges, then the original series is called conditionally (non-absolutely) convergent.
Theorem 1 (a sufficient criterion for the convergence of alternating series)
The alternating series $\sum \limits _(n=1)^(\infty )u_(n) $ converges absolutely if the series composed of the absolute values of its members$\sum \limits _(n=1)^ converges (\infty )\left|u_(n) \right| $.
Comment
Theorem 1 gives only a sufficient condition for the convergence of alternating series . The converse theorem is not true, i.e. if the alternating series $\sum \limits _(n=1)^(\infty )u_(n) $ converges, then it is not necessary that the series composed of modules $\sum \limits _(n=1)^( \infty )\left|u_(n) \right| $ (it can be either convergent or divergent). For example, the series $1-\frac(1)(2) +\frac(1)(3) -\frac(1)(4) +...=\sum \limits _(n=1)^(\infty )\frac((-1)^(n-1) )(n) $ converges according to the Leibniz test, and the series composed of the absolute values of its terms is $\sum \limits _(n=1)^(\infty ) \, \frac(1)(n) $ (harmonic series) diverges.
Property 1
If the series $\sum \limits _(n=1)^(\infty )u_(n) $ converges absolutely, then it converges absolutely for any permutation of its members, and the sum of the series does not depend on the order of the members. If $S"$ is the sum of all its positive terms, and $S""$ is the sum of all absolute values of its negative terms, then the sum of the series is $\sum \limits _(n=1)^(\infty )u_(n) $ is equal to $S=S"-S""$.
Property 2
If the series $\sum \limits _(n=1)^(\infty )u_(n) $ converges absolutely and $C=(\rm const)$, then the series $\sum \limits _(n=1)^ (\infty )C\cdot u_(n) $ also converges absolutely.
Property 3
If the series $\sum \limits _(n=1)^(\infty )u_(n) $ and $\sum \limits _(n=1)^(\infty )v_(n) $ converge absolutely, then the series $\sum \limits _(n=1)^(\infty )(u_(n) \pm v_(n)) $ also converge absolutely.
Property 4 (Riemann's theorem)
If the series conditionally converges, then no matter what number A we take, we can rearrange the terms of this series so that its sum is exactly equal to A; moreover, it is possible to rearrange the terms of a conditionally convergent series in such a way that after that it diverges.
Example 1
Investigate the series for conditional and absolute convergence
\[\sum \limits _(n=1)^(\infty )\frac((-1)^(n) \cdot 9^(n) )(n .\] !}
Solution. This series is sign-alternating, the common term of which we denote: $\frac((-1)^(n) \cdot 9^(n) )(n =u_{n} $. Составим ряд из абсолютных величин $\sum \limits _{n=1}^{\infty }\left|u_{n} \right| =\sum \limits _{n=1}^{\infty }\frac{9^{n} }{n!} $ и применим к нему признак Даламбера. Составим предел $\mathop{\lim }\limits_{n\to \infty } \frac{a_{n+1} }{a_{n} } $, где $a_{n} =\frac{9^{n} }{n!} $, $a_{n+1} =\frac{9^{n+1} }{(n+1)!} $. Проведя преобразования, получаем $\mathop{\lim }\limits_{n\to \infty } \frac{a_{n+1} }{a_{n} } =\mathop{\lim }\limits_{n\to \infty } \frac{9^{n+1} \cdot n!}{(n+1)!\cdot 9^{n} } =\mathop{\lim }\limits_{n\to \infty } \frac{9^{n} \cdot 9\cdot n!}{n!\cdot (n+1)\cdot 9^{n} } =\mathop{\lim }\limits_{n\to \infty } \frac{9}{n+1} =0$. Таким образом, ряд $\sum \limits _{n=1}^{\infty }\left|u_{n} \right| =\sum \limits _{n=1}^{\infty }\frac{9^{n} }{n!} $ сходится, а значит, исходный знакопеременный ряд сходится абсолютно.Ответ: ряд $\sum \limits _{n=1}^{\infty }\frac{(-1)^{n} \cdot 9^{n} }{n!} $ абсолютно сходится.!}
Example 2
Examine the series $\sum \limits _(n=1)^(\infty )\frac((-1)^(n) \cdot \sqrt(n) )(n+1) $ for absolute and conditional convergence.
- We examine the series for absolute convergence. Denote $\frac((-1)^(n) \cdot \sqrt(n) )(n+1) =u_(n) $ and compose a series of absolute values $a_(n) =\left|u_(n ) \right|=\frac(\sqrt(n) )(n+1) $. We get the series $\sum \limits _(n=1)^(\infty )\left|u_(n) \right| =\sum \limits _(n=1)^(\infty )\, \frac(\sqrt(n) )(n+1) $ with positive terms, to which we apply the limit criterion for series comparison. For comparison with $\sum \limits _(n=1)^(\infty )a_(n) =\sum \limits _(n=1)^(\infty )\, \frac(\sqrt(n) )(n+1) $ consider a series that has the form $\sum \limits _(n=1)^(\infty )\, b_(n) =\sum \limits _(n=1)^(\infty )\, \frac(1)(\sqrt(n) ) \, $. This series is a Dirichlet series with exponent $p=\frac(1)(2)
- Next, we examine the original series $\sum \limits _(n=1)^(\infty )\frac((-1)^(n) \cdot \sqrt(n) )(n+1) $ for conditional convergence. To do this, we check the fulfillment of the conditions of the Leibniz test. Condition 1): $u_(n) =(-1)^(n) \cdot a_(n) $, where $a_(n) =\frac(\sqrt(n) )(n+1) >0$ , i.e. this series is alternating. To verify condition 2) on the monotonic decrease of the terms of the series, we use the following method. Consider the auxiliary function $f(x)=\frac(\sqrt(x) )(x+1) $ defined at $x\in )
