S`Q#HĕK2 , miQm` /2 H 7QM+iBQM x2i Hi2`Mû2 /2 _B2KMM

.a 8 , +Q``2+iBQM

o2M/`2/B kk /û+2K#`2 kyRd GǶmb;2 /2b +H+mHi`B+2b MǶ2bi Tb miQ`Bbû

1t2`+B+2 ,

GǶTT`BiBQM /2b i2`K2bp+q 7Bi T2Mb2` ¨ bQKK2` 2M /B;QMH2X

G bû`B2 2bi ¨ i2`K2b TQbBiB7b /QM+ KāK2 Mim`2 [m2 H bû`B2 /2b bQKK2b 2M /B;QMH2bX SQm`n}tû-Sn=P

p+q=nap,q=Pn p=0

1 n+ 1

1

2ⁿ = 1 (n+ 1)2ⁿ

Pn

p=01 = 1 2ⁿX S` i?ûQ`ĕK2 /2 bQKKiBQM 2M /B;QMH2- QM HQ`bP

p

P

qa_p,q=P

nS_n=P

n

1

2ⁿ = 1

1 1/2 = 2T`

bû`B2 ;ûQKûi`B[m2 /2 `BbQM1/2X

1t2`+B+2 ,

RX aQBi">0X S` BMiû;`iBQM T` T`iB2b UH2b 7QM+iBQMb bQMiC¹bm`[",1]-R1

"

arctant

t dt= [ln(t) arctan(t)]¹_"

R1

"

lnt 1 +t²dt.

GQ`b[m2" i2M/ p2`b y-ln(t") arctan(") ⇠"ln(") !0X .QM+ H2 +`Q+?2i ;ûMû`HBbû 2bi +QMp2`;2Mi U/2 HBKBi2 MmHH2V 2i H2b BMiû;`H2b ;ûMû`HBbû2bR1

0

arctant

t dt2iR1 0

lnt

1 +t²dtQMi KāK2 Mim`2- /QM+

+QMp2`;2Mi2b +`R1 0

arctant

t dt2bi 7mbb2K2Mi BKT`QT`2 2M y U+`arctan(t)⇠0t)X PM Q#iB2Mi #B2M H 7Q`KmH2 /2KM/û2X

kX SQm` iQmit2]0,1[- QM t<sup>2</sup>2]0,1[2i /QM+ T` .a1- 1

1 +t² =P+1

n=0( 1)^{n 1}t²ⁿX BMbB- lnt 1 +t² = P₊₁

k=0( 1)^{n 1}t²ⁿlnt=P fn(t)X

G2b 7QM+iBQMbfnBMbB /û}MB2b bQMi +QMiBMm2b T` KQ`+2mt 2i H bû`B2P

fn+QMp2`;2 bBKTH2K2Mi bm`

]0,1[X 1M}M-R1

0|f_n|= 1

(2n+ 1)² [mB 2bi H2 i2`K2 ;ûMû`H /ǶmM2 bû`B2 +QMp2`;2Mi2 T` +QKT`BbQM p2+ H bû`B2 ¨ i2`K2b TQbBiB7bP 1

4n² +QMp2`;2Mi2 T` +`Biĕ`2 /2b bû`B2b /2 _B2KMM U2>1VX 6BMH2K2Mi- T` i?ûQ`ĕK2 /ǶBMiû;`iBQM i2`K2 ¨ i2`K2 /2 "2TTQ@G2pB- QM Q#iB2Mi H 7Q`KmH2 /2@

KM/û2 ,

Z1 0

arctant t dt=

+1X

n=0

( 1)ⁿ (2n+ 1)² jX PM T2mi `2i`Qmp2` +2 `ûbmHii 2M /K2iiMi [m2

8t2R,arctan(t) =

+1X

n=0

( 1)ⁿ 2n+ 1t²ⁿ⁺¹. 2i 2M TTHB[mMi ¨ MQmp2m H2 i?ûQ`ĕK2 /ǶBMiû;`iBQM i2`K2 ¨ i2`K2X

S`Q#HĕK2 , miQm` /2 H 7QM+iBQM x2i Hi2`Mû2 /2 _B2KMM

AX :ûMû`HBiûb RX aQBix 2 Rc bBx > 0- HQ`b H bmBi2Å

1 n^x

ã

n 1

2bi /û+`QBbbMi2 URV p2`b y UkV c /ǶT`ĕb H2 +`Biĕ`2 bTû+BH /2b bû`B2b Hi2`Mû2b- H bû`B2 P

n 1

( 1)^{n 1}

n^x +QMp2`;2 c bBx0- H bmBi2 Å

( 1)^{n 1} n^x

ã

n 1

M2 +QMp2`;2 Tb p2`b y- /QM+ H bû`B2 P

n 1

( 1)^{n 1}

n^x /Bp2`;2 U;`QbbBĕ`2K2MiVX kX *QKK2| t|<1- H bû`B2 ;ûQKûi`B[m2P

( t)ⁿ+QMp2`;2 2i b bQKK2 pmi⁺¹P

k=0

( t)^k= 1 1 ( t) = 1

1 +tc /QM+ H bmBi2(gn)+QMp2`;2 bBKTH2K2Mi p2`b H 7QM+iBQMg:t7! 1

1 +t bm`[0,1[X

• G bmBi2(gn)+QMp2`;2 bBKTH2K2Mi p2`b H 7QM+iBQMgbm`[0,1[c

• H 7QM+iBQMg2i H2b 7QM+iBQMbgn, n2NbQMi +QMiBMm2b UT` KQ`+2mtV c

• +QM/BiBQM /2 /QKBMiBQM,8t2[0,1[, |gn(t)|= 1 ( t)ⁿ⁺¹

1 +t = 1 ( t)ⁿ⁺¹ 1 +t  2

1 +t = (t)c H 7QM+iBQM 2bi BM/ûT2M/Mi2 /2n- +QMiBMm2 UKāK2 bm`[0,1]V 2i BMiû;`#H2 bm`[0,1[X .ǶT`ĕb H2 i?ûQ`ĕK2 /2 +QMp2`;2M+2 /QKBMû2- H bmBi2

ÇZ 1 0

gn

å

+QMp2`;2 p2`bZ1 0

gX P`-Z1

0

gn= Xn

k=0

( 1)^k k+ 1 =

n+1X

k=1

( 1)^{k 1}

k c /QM+F(1) = Z 1

0

g=h

ln(1 +t)i1 0= ln 2X jX 8n 1,8x 2, ( 1)^{n 1}

n^x  1

n²X *QKK2 H bû`B2 P

n 1

1

n² 2bi BM/ûT2M/Mi2 /2n2i +QMp2`;2Mi2- H bû`B2 P

n 1

( 1)^{n 1}

n^x +QMp2`;2 MQ`KH2K2Mi bm`[2,+1[X .QM+ bQM `2bi2Rn(x) = P

n N+1

1

n^x i2M/ p2`b y 2i TQm`N bb2x ;`M/|RN|"/2X 9X S` BHH2m`b-|F(x) 1| P^N

n=2

1 n^x + P

n N+1

1

n^xX GQ`b[m2xi2M/ p2`b+1- H bQKK2 U}MB2V P^N

n=2

1 n^x i2M/ p2`b y T` QTû`iBQMb bm` H2b HBKBi2b +` 1

n^x !0TQm` n 2X AH 2tBbi2 /QM+A >0 i2H [m2 TQm` iQmix > A- P^N

n=2

1 n^x "/2

6BMH2K2Mi- TQm`x > A-|F(x) 1|"/QM+F i2M/ p2`b R HQ`b[m2xi2M/ p2`b+1X 8X GB2M p2+⇣

SQm`x >1- F(x) ⇣(x) = ⁺¹P

n=1

( 1)^{n 1} 1 n^x = ⁺¹P

k=1

2

(2k)^x = 2^{1 x+1}P

k=1

1

k^x = 2^{1 x}⇣(x).PM 2M

/û/mBi HǶû;HBiû ,F(x) = (1 2^{1 x})⇣(x).

eX *QKK22^{1 x} !

x!+1 0-F(x)⇠⇣(x)m pQBbBM;2 /2+12i /QM+⇣(x) !

x!+1 1X dX .û`Bp#BHBiû /2F

X aQBix >0X G 7QM+iBQMh_x:t7!lnt

t^x 2bi /2 +Hbb2C¹ bm`]0,+1[2ih<sup>0</sup><sub>x</sub>(t) = t<sup>x 1</sup>(1 xlnt) t<sup>2x</sup> X .QM+h<sup>0</sup><sub>x</sub>2bi Mû;iBp2 bm` HǶBMi2`pHH2[2<sup>1/x</sup>,+1[2i TQbBiBp2 bm`]0,2^1/x]X .QM+hx2bi /û+`QBbbMi2 bm`[2^1/x,+1[2i +`QBbbMi2 bm`]0,2^1/x]X

PM 2M /û/mBi [m2 H bmBi2Å lnn

n^x ã

n 1

2bi /û+`QBbbMi2 ¨ T`iB` /m `M;b2^1/xc+ 1 [mB 2bi mM 2MiB2` bmTû`B2m` Qm û;H ¨ 2^1/xX

e

#X fn:x7!( 1)^{n 1}2 ^xlnn2bi /2 +Hbb2C¹2if_n⁰(x) = ( 1)ⁿlnn n^x X aQBia >0X PM TQb2N_a =b2^1/ac+ 1X SQm` iQmi x a- H bmBi2Å

lnn n^x

ã

n Na

i2M/ p2`b y 2M /û+`QBbbMi c /QM+ H bû`B2 Hi2`Mû2 P

n Na

f_n⁰(x) +QMp2`;2 2i- TQm`n Na- bQM `2bi2 /ǶQ`/`2n-

⇢n(x)- pû`B}2 ,

|⇢n(x)| ( 1)ⁿ⁺¹ln(n+ 1)

(n+ 1)^x ln(n+ 1) (n+ 1)^a. .QM+ sup

x a|⇢n(x)|  ln(n+ 1)

(n+ 1)^a _n!+1! 0X .QM+ H bû`B2 P

n 1

f_n⁰ +QMp2`;2 mMB7Q`KûK2Mi bm`

[a,+1[X

• SQm` iQmin 1- H 7QM+iBQMf<sub>n</sub>2bi /2 +Hbb2C<sup>1</sup>bm`]0,+1[c

• H bû`B2 P

n 1

f_n+QMp2`;2 bBKTH2K2Mi bm`]0,+1[2i b bQKK2 2biFc

• H bû`B2 P

n 1

f_n⁰ +QMp2`;2 mMB7Q`KûK2Mi bm` iQmi b2;K2Mi BM+Hmb /Mb]0,+1[X .ǶT`ĕb H2 i?ûQ`ĕK2 /2 /û`BpiBQM i2`K2 ¨ i2`K2-F 2bi /2 +Hbb2C¹bm`]0,+1[2i

8x >0, F⁰(x) =

+1X

n=1

( 1)ⁿlnn n^x .

AAX S`Q/mBi /2 *m+?v /2 H bû`B2 Hi2`Mû2 T` 2HH2@KāK2 3X úim/2 /2 H +QMp2`;2M+2

X GQ`b[m2x >1- H bû`B2 P

n 1

( 1)^{n 1}

n^x +QMp2`;2 #bQHmK2Mi c /ǶT`ĕb H2 i?ûQ`ĕK2 /m T`Q/mBi /2

*m+?v UQm bQKK2 2M /B;QMH2bV- H bû`B2 T`Q/mBi /2 P

n 1

( 1)^{n 1}

n^x T` 2HH2@KāK2 +QMp2`;2

#bQHmK2Mi 2i b bQKK2 pmi ,Å₊₁ P

n=1

( 1)^{n 1} n^x

ã2

= (F(x))²X

#X SQm` x > 0- cn(x) = ( 1)^{n 2n}P¹

k=1

1

[k(n k)]^xX *QKK2 k 7! k(n k) 2bi KtBKmK [mM/

k= n

2 2i [m2 H bQKK2 +QKTQ`i2n 1i2`K2b-|cn(x)|=ⁿP¹

k=1

1

[k(n k)]^x (n 1) 1 [(n/2)²]^x = (n 1)4^x

n^2x X SQm` 0 < x  1

2- (n 1)4^x

n^2x mM2 HBKBi2 bi`B+i2K2Mi TQbBiBp2 U}MB2 Qm MQMV- /QM+ H bmBi2 (c<sub>n</sub>(x))M2 +QMp2`;2 Tb p2`b yX .QM+ H bû`B2 P

n 2

c_n(x)/Bp2`;2 ;`QbbBĕ`2K2MiX NX *b Qɍx= 1

X 1

[X] (n [X]) = 1 n

Å 1

[X]+ 1 n [X]

ãX .QM+

cn(1) = ( 1)^{n 2}

n 1

X

k=1

1

k(n k) = ( 1)^{n 2}1 n

nX1 k=1

Å1 k + 1

n k

ã

= ( 1)^{n 2}1 n

n 1

X

k=1

1 k+

n 1

X

k=1

1 n k

!

= 2( 1)^{n 2}1 n

nX1 k=1

1

k = 2( 1)^{n 2}Hn 1

n .

#X JQMQiQMB2 Hn 1

n

Hn

n+ 1 = 1 n

Å Hn 1

n

ã Hn

n+ 1=Hn

Å1 n

1 n+ 1

ã 1 n² Å

1 +1 2

ã 1 n(n+ 1)

1

n² = n 2 2n²(n+ 1) 0.

.QM+ H bmBi2Å H_{n 1}

n ã

n 2

2bi /û+`QBbbMi2X

+X S` +QKT`BbQM bû`B2bfBMiû;`H2b p2+ H 7QM+iBQMt7!1/t- QM T2mi ûi#HB` H2 `ûbmHii +HbbB[m2 , Hn⇠lnnm pQBbBM;2 /2+1X .QM+ H bmBi2ÅH_{n 1}

n ã

n 2

+QMp2`;2 p2`b y 2M /û+`QBbbMi 2i H bû`B2 Hi2`Mû2 P

n 2

c_n(1)+QMp2`;2X

AAAX *H+mH /2 H bQKK2 /ǶmM2 bû`B2 ¨ HǶB/2 /ǶmM2 ûim/2 /2⇣ m pQBbBM;2 /21 RyX .ûp2HQTT2K2Mi bvKTiQiB[m2 2M R

X PM TQb2h=x 1X *QKK2F 2bi /û`Bp#H2 2M R- m pQBbBM;2 /21- QM , F(x) =F(1) +hF⁰(1) +o(h) = ln 2 +hF⁰(1) +o(h).

PM mbbB ,1 2^{1 x}= 1 2 ^{hln 2}=hln 2 ln²2

2 h²+o(h²)m pQBbBM;2 /2x= 1X

#X .ûp2HQTT2K2Mi /2⇣

⇣(x) = F(x)

1 2^{1 x} = ln 2 +hF⁰(1) +o(h) hln 2 ln²2

2 h²+o(h²)

= 1

hln 2

ln 2 +hF⁰(1) +o(h) 1 ln 2

2 h+o(h)

= 1

hln 2(ln 2 +hF⁰(1) +o(h)) Å

1 +ln 2

2 h+o(h) ã

= 1

hln 2 Ç

ln 2 +h Ç

F⁰(1) +ln²2 2

å +o(h)

å

= 1

h+ ÅF⁰(1)

ln 2 +ln 2 2

ã +o(1) RRX .ûp2HQTT2K2Mi bvKTiQiB[m2 2M R U#BbV

X SQm`n 12ix2[1,2]-t7! 1

t^x 2bi /û+`QBbbMi2 bm`[n, n+ 1]U[mB 2bi mM BMi2`pHH2 /2 HQM;m2m`

RV- /QM+1. 1 (n+ 1)^x 

Z n+1 n

/t t^x 1. 1

n^xX PM 2M /û/mBi [m2 ,0vn(x) 1 n^x

1 (n+ 1)^x.

#X SQm` x 2 [1,2]- H bmBi2 Å 1 n^x

ã

n 1

+QMp2`;2 Up2`b yV c +QKK2 Pⁿ

k=1

Å 1 k^x

1 (k+ 1)^x

ã

= 1 1

(n+ 1)^x- H bû`B2 P

n 1

Å 1 n^x

1 (n+ 1)^x

ã+QMp2`;2X .2 HǶ2M+/`2K2Mi /m UV- QM /û/mBi H +QMp2`@

;2M+2 /2 H bû`B2 P

n 1

vn(x)X +X SQm`x2]1,2]- Pⁿ

k=1

vk(x) = Pⁿ

k=1

1 k^x

Z n+1 1

/t

t^x _n!+1!⇣(x) Z +1

1

/t

t^x =⇣(x) 1 x 1X /X G bû`B2 P

n 1

vn +QMp2`;2 bBKTH2K2Mi bm`[1,2]X PM MQi2 Rn(x) = ⁺¹P

k=n+1

vk(x)H2 `2bi2 /ǶQ`/`2 n/2 H bû`B2X .ǶT`ĕb UV-0Rn(x) ⁺¹P

k=n+1

Å 1 k^x

1 (k+ 1)^x

ã

= 1

(n+ 1)^x lim_k!+1 1 k^x = 1

(n+ 1)^xX .QM+ sup

x2[1,2]|Rn(x)| 1

(n+ 1)¹ _n!+1!0X .QM+ H bû`B2 P

n 1

vn +QMp2`;2 mMB7Q`Kû@

K2Mi bm`[1,2]X

3

2X SQm`x2]1,2]-v_n(x) = 1 n^x

1 1 x

Å 1 n^{x 1}

1 (n+ 1)^{x 1}

ãcv_n(1) = 1

n ln(n+ 1) + lnnX vn 2bi +QMiBMm2- bm7 T2mi@āi`2 2M RX

1M R , 2M TQbMih=x 1- 1 n^x = 1

n+o(1)T` +QMiBMmBiû /2 HǶ2tTQM2MiB2HH2x7!n ^x 2M R 2i 1

1 x Å 1

n^{x 1}

1 (n+ 1)^{x 1}

ã

= 1

h 2 ^hlnn 2 ^hln(n+1) = 1 h

⇣(1 hlnn+o(h)) (1 hln(n+

1) +o(h)⌘

= ln(n+ 1) lnn+o(1)c /QM+ vn(x) = 1

n+ ln(n+ 1) lnn+o(1)X .QM+vn2bi +QMiBMm2 2M RX

PM 2M /û/mBi [m2 H bû`B2 P

n 1

v_n2bi mM2 bû`B2 /2 7QM+iBQMb +QMiBMm2b bm`[1,2]X G +QMp2`;2M+2 mMB7Q`K2 bm`[1,2]2Mi`ŗM2 /QM+ H +QMiBMmBiû /2 b bQKK2 bm`[1,2]X

PM 2M /û/mBi [m2⇣(x) 1

x 1 =⁺¹P

n=1

vn(x) = Å₊₁

P

n=1

vn(1) ã

+o(1) = +o(1)m pQBbBM;2 /2 1⁺X .ǶQɍ⇣(x) = 1

x 1+ +o(1)m pQBbBM;2 /21⁺X RkX TTHB+iBQM

S` mMB+Biû /m /ûp2HQTT2K2Mi HBKBiû 2M1<sup>+</sup>Uûp2Mim2HH2K2Mi 2M KmHiBTHBMi T`(x 1)V- QM /û/mBi /2 3XU#V 2i NXU2V H2b û;HBiûba= 12i F⁰(1)

ln 2 +ln 2

2 =b= X .ǶQɍF⁰(1) = ln 2

Å ln 2 2

ãX

.ǶT`ĕbAX9XU#V-⁺¹P

n=1

( 1)^{n 1}lnn

n = F⁰(1) = ln 2 Åln 2

2 ãX

AoX *H+mH /2 ⇣(4)2i TTHB+iBQM ¨ H T?vbB[m2 , SQm`x >1- QM TQb2⇣(x) =X

n 1

1

n^xX PM `TT2HH2 U2i QM /K2iV [m2⇣(2) =X

n 1

1 n² =⇡²

6 . SQm`m2N^⇤2in2N^⇤- QM TQb2 , (m, n) = 2

mn³+ 1 m²n²+ 2

m³nX

RjX X aQBi X

m 1

X

n 1

(m, n) X

m 1

X

n 1

(m, n+m) X

m 1

X

n 1

(m+n, n).

.ǶmM2 T`i- QM T2mi pû`B}2` [m2 (m, n) (m, n+m) (m+n, n) = 2 m<sup>2</sup>n<sup>2</sup>X HQ`b

X

m 1

X

n 1

(m, n) X

m 1

X

n 1

(m, n+m) X

m 1

X

n 1

(m+n, n) = X

m 1,n 1

2 m²n²

= X

m 1

ÑX

n 1

2 m²n²

é

=X

m 1

2 m²

ÑX

n 1

1 n²

é

= 2 ÑX

n 1

1 n²

é ÑX

m 1

1 m²

é

= 2⇣(2)². S` BHH2m`b ,

G T`2KBĕ`2 bû`B2 +Q``2bTQM/ ¨ H bQKK2 /2 iQmb H2b i2`K2b (a, b)TQm`(a, b)2(N<sup>⇤</sup>)<sup>2</sup>UTHM bmTû`B2m` /`QBiVX

G b2+QM/2 bû`B2 +Q``2bTQM/ ¨ H bQKK2 /2b i2`K2b (a, b)i2Hb [m2(a < b)bBimûb bi`B+i2K2Mi m /2bbmb /2 H /B;QMH2 /m [m`i /2 THMX

G i`QBbBĕK2 bû`B2 +Q``2bTQM/ ¨ H bQKK2 /2b i2`K2b (a, b)i2Hb [m2(a > b)bBimûb bi`B+i2K2Mi 2M /2bbQmb /2 H /B;QMH2 /m [m`i /2 THMX

6BMH2K2Mi- HǶ2Mb2K#H2 +Q``2bTQM/ ¨ H bQKK2 /2b i2`K2b (a, b)i2Hb [m2a=bUH /B;QMH2 /m THMV- +Ƕ2bi ¨ /B`2P

a2N (a, a) =P

n 1

5

a⁴ = 5·⇣(4)X

#X .QM+5⇣(4) = 2⇣(2)²= 2⇡⁴

36 2i QM T2mi HQ`b 2M /û/mB`2 [m2

⇣(4) =X

n 1

1 n⁴ =⇡⁴

90.

R9X .ûi2`KBM2`- T`ĕb pQB` DmbiB}û bQM 2tBbi2M+2- 2i 2M /ûiBHHMi H2b +H+mHb HǶBMiû;`H2Z +1 0

t³ 2^t 1dt SQm`t >0- +QKK20exp( t)<1- QM t³

e^t 1=t³e ^t 1

1 e ^t =t³P

n2Ne ^(n+1)t=P fn(t)X

*?[m2fn2bi +QMiBMm2 bm`]0,+1[- TQbBiBp2b 2i BMiû;`#H2 +` 7mbb2K2Mi BKT`QT`2 2M y 2io(1/t<sup>2</sup>) 2M+1X G bû`B2 /2 7QM+iBQM +QMp2`;2 bBKTH2K2Mi p2`bf [mB 2bi +QMiBMm2 2i BMiû;`#H2X

SQm`n2N-Z

|fn|= 1 (n+ 1)⁴

Z +1 0

u³exp( u) du2M TQbMiu= (n+1)t+?M;2K2Mi /2 p`B#H2b

#BD2+iB7X

S` ASS Qm 2M `2+QMMBbbMi H 7QM+iBQM - QM }MH2K2Mi Z

|fn| = 6

(n+ 1)⁴ [mB 2bi H2 i2`K2

;ûMû`H /ǶmM bû`B2 +QMp2`;2Mi2 T` +`Biĕ`2 /2 _B2KMMX .QM+ T` BMiû;`iBQM i2`K2 ¨ i2`K2 /2 "2TTQ@G2pB-Z

f=X

n 0

6

(n+ 1)⁴= 6⇣(4) = ⇡⁴ 15X R8X aQBiM = c

4 Z +1

0

8⇡hc

5

1

exp(_kB^hc_T) 1d X PM TQb2t='( ) = hc

k_B T +?M;2K2Mi /2 p`B#H2b #BD2+iB7X HQ`b = hc

kBtT 2id = hc kBt²T dt 1iM =

Z 0 +1

2⇡hc²k_B⁵t⁵T⁵ h⁵c⁵

( hc) kBt²T

1

exp(t) 1dt= Z ₊₁

0

2⇡hc²k_B⁵t⁵T⁵ h⁵c⁵

hc kBt²T

1 exp(t) 1dt T`ĕb bBKTHB}+iBQMb 2i /ǶT`ĕb HǶBMiû;`H2 +H+mHû2 H [m2biBQM T`û+û/2Mi2-

...= 2⇡k_B⁴T⁴ h³c²

⇡⁴ 15= T⁴ p2+ = 2⇡⁵k⁴_B

15h³c²X

Ry