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Higher Mathematics. Part 3

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∞

n −1

1.6.13. ∑

n=1 n

+ n +

∞

− 1

1.6.16. ∑

n=1

∞

1.6.19. ∑ arcsin

+ 1

n=1

∞

n + 1

1.6.22. ∑

n + 3

n=1

∞

1.6.25. ∑arctg

+ 2

n=1

−

∞

− e

1.6.28. ∑

n=1

∞

1.6.14. ∑

− cos

n=1

∞

2n sin

1.6.17. ∑

n=1

3n + 2

∞

1.6.20. ∑

tg3

n=1 n

4 n

∞

2n+1

1.6.23. ∑

n=1

∞1

1.6.26.n∑=1 n2 5n .

∞	1		n + 1
1.6.29. ∑		ln		.
n=1 n			n
n=1 n			n

∞

1.6.15. ∑sin3

n=1

(n+1)(n+2)

∞

n − 2

1.6.18. ∑

n=1 n

+ 4

∞

n + 2

1.6.21. ∑ln

n=1

∞

1.6.24. ∑

sin4

n=1

7 n

∞

1.6.27. ∑

− cos

n=1

∞en3 − 1

1.6.30.∑ 1 .=

n 1 sin n

1.7. Using the integral test or the comparison test, examine the series for convergence.

∞

1.7.1. ∑

n=1 n ln2

(n + 4)

∞

1.7.3. ∑

n=3 n ln n ln(ln n)

∞

1.7.5. ∑

+ 1) ln(n +

n=1

∞

sin

1.7.7. ∑

(n + 1)

n=1 ln

∞

1.7.9. ∑

n=2

ln n

∞

1.7.11. ∑

n=2 n(1+ ln2

∞

ln n

1.7.2. ∑

n= 2 n(ln2

n + 1)

∞

1.7.4. ∑

n=3 n ln n ln2

(ln n)

∞

1.7.6. ∑

(2n2

+ 3) ln4

n=2

∞

1.7.8. ∑

n=1 ln(n

∞

1.7.10. ∑ ne− n2 .

n=1

∞

ln n

1.7.12. ∑

+ ln4

n=2 n(4

∞	arctg	1
		n
1.7.13. ∑			.

n=2	3 ln n

1.7.15. ∞ n 3− n2 .

∑

n=1

∞1

1.7.17.n∑=1 n ln6 (n + 2) .

∞3

1.7.19.∑ n2 e−n .

n=1
∞
1.7.21. ∑ ne−n						n .
n=1
∞				1
1.7.23. ∑				1					.
1.7.23. ∑									.
n=2 n(9 + 4 ln2							n)
∞		−		n+1
1.7.25. ∑	3					.
1.7.25. ∑			n + 1			.
n=1			n + 1
∞				n2 + 1
1.7.27. ∑								.
1.7.27. ∑				ln2				.
n=1 n3				ln2		(n + 1)
∞	2		−	n
1.7.29. ∑	2				.
1.7.29. ∑					.
n=1				n

∞1

1.7.14.n∑=1 n ln3 (2n + 1) .

∞1

1.7.16.∑ .2

n=2 n(4 + ln n)

∞1

1.7.18.n∑=1 (n + 2) ln(n + 5) .

∞				1
1.7.20. ∑				1				.
1.7.20. ∑	(n + 3) ln3							.
n=1	(n + 3) ln3				(n + 4)
∞				n2
1.7.22. ∑							.
1.7.22. ∑	(n		3				.
n=1	(n			+ 2) ln(n + 5)
∞
1.7.24. ∑ n3e−n4 .
n=1
∞
1.7.26. ∑ n 5−n2 .
n=1
∞				ln n
1.7.28. ∑				ln n		.
1.7.28. ∑						.
n=2		n(1+ ln4 n)

∞n + 2

1.7.30.n∑=1 (n2 + 4) ln(n + 1) .

1.8. Examine for conditional convergence and absolute convergence the series.

∞

(−1)

n+1

∞

(−1)

n+1

(2n + 3)

1.8.1. а)

∑

;

∑

n ln n

n=2

n=1

n3 + 1

∞

(−1)

n+1

∞

(−1)

n+1

(n − 1)

1.8.2. а)

∑

;

∑

3n − 1

n=1

4n+ 2

∞

(−1)n+1 tg

∞

(−1)

n+1

+ 1) .

1.8.3. а)

∑

;

∑

n=1

n n

n=1

n4 + 1

∞

(−1)

n ;

∞

(−1)

n+1

1.8.4. а)

∑

n=1

5n2 + 1

n=1

∞(−1)n+1

1.8.5.а) n∑=1 (n + 1) 2n+1 ;

	∞	(−1)n				2n − 1		n
1.8.6. а)	∑					2n − 1		;
1.8.6. а)	∑							;
	n=1	( )			3n + 2
	∞	−1		n	(n + 1)
1.8.7. а)	∑	−1			(n + 1)		;

		3n	2	− 7n + 1
	n=1	3n		− 7n + 1

∞(−1)n+1

1.8.8.а) n∑=1 (2n + 1) ln2 (2n + 1) ;

∞(−1)n+1 n2

1.8.9.а) ∑ ;n

n=1

∞

(−1)n+1 sin

1.8.10. а)

∑

;

n=1

∞

(

−1)

n+1

1.8.11. а)

∑

;

n=1

∞

(−1)

1.8.12. а)

∑

;

3 5n − 3

n=1

∞

(−1)n+1 arctg

1.8.13. а)

∑

;

n=1

∞

(

−1)

2n + 1

1.8.14. а)

∑

;

n=1

2n + 3

∞

(−1)

n+1

1.8.15. а)

∑

;

(

2n −

)

n=1

∞

(−1)n

n +

1.8.16. а)

∑

;

n=1

3n − 2

∞

(−1)

(3n − 1)

1.8.17. а)

∑

;

n=1

5n2 + 7

∞

(−1)n+1 sin

1.8.18. а)

∑

;

n=1

b) ∑ (−1)

n+1

∞

n=2 n ln n

∞

(−1)

n+1

(4n + 1) .

b) ∑

n=1

n(3n − 1)

b) ∑ (−1)

n+1

∞

n=2

n ln3 n

∞

(−1)(3n + 1)

b) ∑

n=1

n2 + 1

∞

(−1) (2n + 1)

b) ∑

n=1

n(5n − 1)

∞

(−2)

∑

n=1

(n + 2)!

b) ∑ (−1)

n+1

∞

n=1 5 n2 + 7

∞

(−1)

n+1

n−1

∑

n=1

(n + 1)!

∞

(−1)

n+1

(2n − 1)

b) ∑

n=1

b) ∑ (−1)

n+1

∞

n=1 3 4n2 + 1

∑

(

)

n+1

∞

−1

n=1 3 n2

n + 1

∑ (−1)

∞

n=2 n 3 ln n

∞

(−1)

b) ∑

5n−2

n=1

∞

(−1)n

n − 1

b) ∑

n=1

n + 1

1.8.19. а)	∞	(−1)n n 2n	;
1.8.19. а)	∑	3n	;
	n=1	3n
	∞	(−1)n arcsin		1
1.8.20. а)	∑			1	;
1.8.20. а)	∑			4n + 1	;
	n=1			4n + 1
	n=1

∞(−1)n n

1.8.21.а) n∑=1 2n2 + 3 ;

∞

(−1)n sin

1.8.22. а)

∑

;

n=1

∞

3n2

+ 1

1.8.23. а)

∑

(−1)

;

n=1

− 1

∞

(−1)n tg

1.8.24. а)

∑

;

n=1

∞

(−1)

n + 1

1.8.25. а)

∑

;

n 1

n + 3

∞

(−1)n arctgn

3n −1

1.8.26. а)

∑

;

n=1

3n2 +1

1.8.27. а)

∞

(−1)

n+1

;

∑

n=1

n ln n

∞

(−21) n ;

1.8.28. а)

∑

n=1

n + 4

∞

(−1)n+1

+ 2

1.8.29. а)

∑

;

n=1

5n3 − 1

∞

(−1)n arctg

1.8.30. а)

∑

;

n 3

n 1

∑ (

−1)

n+1

∞

n=1

3n + 4

∞

(−1) (2n − 1)

b) ∑

n=1

5n3 + 9

∑ (−1) .

∞

n=2

n ln2 n

∑ (

−1)

n+1

∞

n=1 3 n2 + 3

b) ∑ (−1)

n+1

∞

n=1 4 n3 + 1

∞

(−1)

n n − 2 n2

∑

n 1

n +

∞

3n + 1 n

∑

−

n 1

7n − 1

b) ∑ (−1)

∞

n=1

n2 + 1

∞

(−1)

n+1

(2n + 1)

b) ∑

n=1

∞

(

−1)

n+1

n+4

∑

n=1

b) ∑ (−1)

n+1

∞

n=2 n ln4 n

∞

(−1)

n+1

(5n − 1)

b) ∑

n=1

5n2 + 1

Micromodule 2

BASIC THEORETICAL INFORMATION. FUNCTIONAL SERIES

Functional series. General definitions. Uniform convergence. Weierstrass’ test. Properties of uniformly convergent series. Power series. Abel’s theorem. Interval and radius of convergence of a power series. Taylor’s and Maclaurin’s series. Expansion of a function into a series. Applications of the series.

Key words: domain of convergence — область збіжності, alternating series —

знакопереміжний ряд, plus-and-minus series — знакозмінний ряд, dominated series — мажорований ряд, dominating series — мажоруючий ряд, expansion — розклад, remainder — залишок, term-by-term integration — почленне інтегрування, absolutely convergent — абсолютно збіжний, termwise integration — почленне інтегрування, Maclaurin’s series — ряд Маклорена, Taylor’s series — ряд Тейлора, uniform convergence — рівномірна збіжність, Abel’s theorem —

теорема Абеля, power series — степеневий ряд.

Literature: [3, chapter 5, sections 5.4 — 5.5], [9, chapter 9, § 2], [14, chapter 3, § 2], [15, chapter 13, sections 13.2—13.3], [16, chapter 16, § 9—28], [17, chapter 5, § 16—19].

2.1. General Definitions

Suppose {un (x)} = {u1 (x), u2 (x), …, un (x), …} is a sequence of functions defined on some domain D.

∞

Definition. ∑ un is called a functional series if all terms of this series are

n=1

functions of some variable x.

∞

u1 (x) + u2 (x) + ... + un (x) + ... = ∑ un (x) (2.1)

n=1

Assigning to x different numerical values we get different numerical series which may prove to be convergent or divergent.

Definition. The set of all values of x for which the functional series converges is called the domain of convergence of the series.

Definition. A sum

Sn (x) = u1 (x) + u2 (x) +…+ un (x)

of the first n terms of series (2.1) is called the п-th partial sum of this series .

There exists lim Sn (x) = S(x) at all points of a domain of convergence.

n→∞

This limit is called a sum of a series (2.1).

In the domain of convergence of a functional series its sum is also a function of x.

Definition. If a functional series (2.1) is convergent then a difference rn (x) = S(x) − Sn (x) is called the п-th remainder of a series:

rn (x) = un+1 (x) + un+ 2 (x) + … .

lim rn (x) = 0 at all points of a domain of convergence.

n→∞

Definition. A functional series (2.1) is called absolutely convergent if

∞

| u1(x) | + | u2 (x) | +...+ | un (x) | +... = ∑| un (x) | is convergent.

n=1

To find a domain of convergence of functional series we may use tests of convergence of numerical series. For example, using d’Alembert’s test we find

lim	un+1(x)	= l(x) . Then we solve inequality l(x) < 1 and investigate points for
	un (x)
n→∞

which l(x) = 1. By analogy we may use Cauchy’s test.

2.2. Uniform Convergence of Functional Series

Definition. The functional series u1(x) + u2 (x) + …+ un (x)… is called domi-

nated in some range of x if a convergent numerical series with positive terms

α1 + α2 + α3 +…+ αn +… such as x from this range |u1(x)| ≤ α1; |u2(x)| ≤ ≤ α2; …|un(x)| ≤ αn, ….

Theorem	Let functional series u1(x) + u2 (x) + …+ un (x)… be dominated
	on the segment [a; b]. Let S(x) be the sum of this series and
Sn (x) be the sum of the first n terms of this series.

Then ε > 0 there will be a positive integer N such as n > N the following inequality is true:

S(x) − Sn (x) < ε x [a; b].

Definition. A functional series (2.1) is called a uniformly convergent series on the set D, if for any small number ε > 0 there exists such

number N = N(ε), that for all n > N and for all x D the following inequality is true

rn (x) < ε.

Thus, a functional series u1(x) + u2 (x) + …+ un (x)… is called a uniformly convergent if ε > 0 N such as n > N:

S(x) − Sn (x) <ε x [a; b].

The dominated series is a series of uniformly convergence.

Let’s consider geometrical interpretation of a uniform convergence of a functional series. Suppose a functional series (2.1) is uniformly convergent on

(a; b),	S(x) is its sum, Sn (x) is a partial sum of this series. Let’s consider for
any ε > 0 the graphs of the functions y = S(x),		y = S(x) + ε and y = S(x) − ε on
(a; b)	(Fig. 2.1, а). Graphs of two last functions form a band 2ε in width. If the
series	(2.1) uniformly converges on (a; b)	to the function	S(x) , then there
exists such a number N = N(ε), that graphs of all partial sums			y = Sn (x), n > N
on (a; b) are within 2ε -band.

For non-uniformly convergent series such number N does not exist (Fig. 2.1, b).

y	y = S(x) + ε		y	y = S(x) + ε
	y = S(x) + ε			y = S(x) + ε
	y = S(x)	2ε		y = S(x)	2ε
	y = S(x)			y = S(x)
	y = Sn(x)			y = Sn(x)
	y = S(x) – ε			y = S(x) – ε
О a	b	x	О a	b	x
	а			b

Fig. 2.1

To investigate functional series for uniform convergence we may use sufficient test of uniform convergence.

Theorem	(Weierstrass’ test) A functional series (2.1) is absolutely and
	uniformly convergent on a set D, if there exists a convergent

	∞
numerical series ∑ an with such positive terms that for all		x D the following
	n=1
inequalities are true
	\| un (x) \|≤ an ( n = 1, 2, …).
	(Cauchy’s test). A functional series (2.1) uniformly converges on
Theorem
	a set D if and only if for any ε > 0 there exists	N = N(ε) such as for
	a set D if and only if for any ε > 0 there exists	N = N(ε) such as for

all n > N , for any natural number p and for all x D the following inequality is true

Sn+ p (x) − Sn (x) < ε .

n=1α

2.3. Continuity of a Sum of Functional Series

Let’s consider a series u1(x) + u2 (x) + …+ un (x)… which is convergent on some interval [a; b].

		The sum of the series of continuous functions dominated on
	Theorem	The sum of the series of continuous functions dominated on
	Theorem	some interval [a; b] is a continuous function on this interval.

The converse statement isn’t true. There are series not dominated on an interval, which converge on that interval to the continuous functions.

Every uniformly convergent series on an interval [a; b] has a continuous function for its sum if all terms of this series are continuous functions.

2.4. Integration and Differentiation of Series

Let u1(x) + u2 (x) +…+ un (x)… be a dominated series on the interval [a; b].

Let S(x) be the sum of this series. Then the integral of S(x) from α to x (α [a; b] and x [a; b]) is equal to the sum of the integrals of terms of the given series.

∞

S(x) = ∑ un (x)

n=1

x	x	x
= ∫ u1	(t )dt + ∫ u2	(t )dt + ... + ∫ un
α	α	α

∫x S (t )dt =

∞ x

(t )dt + ... = ∑ ∫ un (t)dt.

Remark. Suppose a series isn’t dominated. Then term-by-term integration of it isn’t always equal to the sum of the integrals of terms of this series.

Theorem	Suppose a series		u1(x) + u2 (x) + …+ un (x)… (where un(x) are
	continuous	functions) converges		to	the sum	S(x). Functions
un (x) have	derivatives	′	n and series	′	′	′
		un (x)		u1 (x) + u2 (x) + …+ un (x)… is

dominated on the interval I. Then the sum of derivatives is equal to the derivative of the sum of the given series.

S′(x) =	∞	′	∞
	∑un (x)		= ∑un′ (x), x I.
n=1			n=1

Remark. The requirement of dominance of a series of derivatives is very essential. If it is not met it makes term-by-term differentiation of the series impossible.

2.5. Power Series

Definition. A functional series

∞
a0 + a1x + a2 x2 + ...+ an xn + ... = ∑ an xn .	(2.2)
n=0

(where a0, a1, …, an,… are constants) are called a power series. a0, a1, …, an are called the coefficients of a power series.

The domain of convergence of a power series is an interval, which can degenerate into a point.

We may also consider a power series in powers of x − x0 , that is

a0	+ a1(x − x0 ) + a2	∞
		(x − x0 )2 + ...+ an (x − x0 )n + ... = ∑ an (x − x0 )n , (2.3)
		n=0
where	x0 is any constant. If we substitute x − x0 = t we get a series (2.2).

Domain of convergenсe of a power series (2.2) contains at least one point x = 0 (a series (2.3) converges at point x = x0 ).

Theorem	(Abel’s theorem):

1)If a power series converges for some value x = x1 ≠ 0, => it converges absolutely for x: | x |< | x1 | ;

2)If a power series diverges for some value x = x2 ≠ 0, => it diverges x

for such as | x |>| x2 | . (Fig 2.2).

A series

–х1

Convergentseries

х1

is divergent

–х2

х2

Fig. 2.2

∞

converges. Then lim an x1n = 0 in accordance

Proof. A power series ∑ an x1n

n=1

→∞

a xn

with necessary condition of convergence. Therefore

is bounded. That is

there exists such a number

M >

is fulfilled:

0 that the following

inequality

an x1n

≤ M , n = 0, 1, 2, … .

Suppose | x |< | x | .

Let q =

, then q < 1 and

an xn

an x1n

≤ M qn .

Then for all x such as |

x |< |

x1 | the series (2.2) is absolutely convergent

according to the comparison test.

From this theorem it may be concluded that there exists a number R such as for |x| < R the series converges; for |x| > R it diverges.

	Theorem	The domain of converges of a power series is an interval with the
	Theorem	center at the origin.
		center at the origin.

Definition. The interval of convergence of a power series is an interval from — R to R. x lying inside this interval a series converges and points x outside the interval a series diverges.

Abel theorem characterizes sets of points of convergence and divergence of power series. The following cases are possible:

1)a series is convergent only at one point x = 0 ;

2)a series is convergent for all x (−∞; ∞) ;

3) R > 0 that for | x |< R a series is absolutely convergent and for | x |> R it is divergent (Fig. 2.3).

Series	Series is		Series
is divergent	convergent		is divergent
–R	0	R	х
	Fig. 2.3

Definition. A number R is called a radius of convergence of power series. Table 2.1 contains a connection between a radius of convergence and interval of convergence of power series (2.2), (2.3).

		Table 2.1

Radius of convergence R	Interval of convergence	Interval of convergence
Radius of convergence R	of a power series (2.2)	of a power series (2.3)
	of a power series (2.2)	of a power series (2.3)

R = 0	x = 0	x = x0
R = ∞	(−∞; ∞)	(−∞; ∞)

0 < R < ∞	(−R; R)	(−R + x0 ; R + x0 )

Let’s find a radius of convergence. We consider a series formed as modules of terms of power series (2.2)

| a0 | + | a1x | + | a2 x2 | +...+ | an xn | +...

Let’s use d’Alembert’s test. Suppose there exists the limit

n+1

xn+1

= | x | lim

≠ 0 , x ≠ 0.

lim

= lim

a xn

n→∞

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