rvn` ogea oexzt ∗

aygnd ircne dpkez zqcpdl 2 `"ecg

1 dl`y mi`ad milxbhpi`d z` eayg .1

Z

"

1

#

u = arctan(x) du = x2dx +1 2 v = x2 dv = xdx 1 Z 1 1 x2 dx x2 − = arctan(x) 2 2 −1 x2 + 1 x=−1 √ √ Z 1 2 Z 1 2 − 2 x +1 1 1 1 1 = ( − )− dx + dx 2 2 2 2 −1 x2 + 1 2 −1 x2 + 1 √ 1 √ 2 1 − 1 + arctan(x) = 2−1 = 2 2

x arctan(x)dx = −1

x=−1

.2

1

Z

√

0

1

  x t−5 √ dt dx = t = 5 − 4x, dt = −4dx = 5 − 4x 5 16 t Z 5  Z 5 1 = 5t−1/2 dt − t1/2 dt 16 1 1 √   5 1 5 5 + 17 3 3/2 1/2 = = 10t − t 16 2 32 t=1 Z

2 dl`y mi`ad milxbhpi`d ly i`pzae hlgda zeqpkzd exwg

2 `ed oezpd lxbhpi`d hxtae ,( , ∞) megza π

ziaeig :

. lim

x→∞

R∞ 1 1 x

Z `id

f (x) = sin(1/x)

divwpetd ik al miyp :

1

∞

1 sin x

  1 dx x

.1

dx x2 qpkznd lxbhpi`d mr zileab d`eeyd lxbhpi`d lr rvap .iaeig

sin 1 x2

1 x




  1 = lim x sin =1 x→∞ x

.(hlgda) qpkzn oezpd lxbhpi`d hxta .cgia mixcazne miqpkzn milxbhpi`d okle ,0-n lecbd iynn leab eplaiw

Z :ze`ad zeivwpetd lk ik al miyp :

1

x 7→ 2x + 5, x 7→ x + 1, x 7→ arctan(x), x 7→

√

∞

x + 1 arctan(x) √ dx · 2x + 5 x

.2

x

d`eeyd rvap .(hlgda qpkzn m"m` qpkzn ok lr) iaeig `ed oezpd lxbhpi`d hxtae ,(0, ∞) megza zeiaeig od R ∞ dx √ xcaznd lxbhpi`d mr zileab : 1 x

. lim

x→∞

x+1 2x+5

·

arctan(x) √ x √1 x

= lim

x→∞

x+1 π · lim arctan(x) = x→∞ 2x + 5 4

oeeikn .qpkzn oezpd lxbhpi`d xnelk ,cgia mixcazne miqpkzn milxbhpi`d okle ,0-n lecbd iynn leab eplaiw .i`pza zeqpkzd oi` ,ziaeig divwpetdy ∗

elawzd dl` mbe oexztl zexg` zehiya mb ynzydl ozip milibxzdn wlga .ogeal miixyt` zepexzt mb mibven

1

2

3 dl`y :`ad ixtqnd xehd zeqpkzd egiked .1

∞ X sin(n2 + 1) arctan(3n + 1) (n + 1)[ln(n + 1)]3 n=1

d`eydd ogan z` dilr rval ozip `l ok lre ,zipehepen dpi`e ziaeig dppi` df dxwna xehd ixai` zxcq

n∈N

ik miiwzn

•

lkl ,oiicr .lxbhpi`l

sin(n2 + 1) arctan(3n + 1) π 1 ≤ 3 (n + 1)[ln(n + 1)] 2 (n + 1)[ln(n + 1)]3 P∞ 1

ok lre) hlgda qpkzn oezpd xehd ik lawp f`e ,qpkzn

n=1 (n+1)[ln(n+1)]3 xehd ik `ceel epl witqny jk .(qpkzn

1 → ∞ xy`k 0-l zt`ey ,ziaeig `id (n+1)[ln(n+1)] 3 dxcqd ik al miyp • P∞ dx 1 lxbhpi`d m"m` qpkzn n=1 (n+1)[ln(3n+1)]3 xehd okl .(dxiyi (x + 1)[ln(x + 1)]3 1 Z ∞ Z ∞   dx dt dx = t = ln(x + 1), dt = x+1 = 3 3 1)[ln(x + 1)] (x + 1 ln(2) t

dwica i"r) zipehepen dxcqd ike ,n Z ∞ ok enk .qpkzn

.qpkzn oezpd xehd okle ,qpkzn `ed

P∞

.

n=1

ln(n) n n2 x zewfgd xeh ly zeqpkzdd megz z` eayg .2

iyew zgqep i"r zeqpkzdd qeicx z` aygp ziy`x

•

r 1 n ln(n) . = lim R n→∞ n2 ≤ ln(n) ≤ n

ok lr .1

r 1 = lim

n→∞

n

1 ≤ lim n→∞ n2

r n

r

ln(n) ≤ lim n→∞ n2

n

ik miiwzn

n∈N

lkl ik al miyp

n =1 n2

.(−1, 1) megza qpkzn xehde ,R

=1

-uiacpq llk i"tr ,hxtae

:zeevwa zeqpkzd zrk wecap

n ∈ N

lkl ik al miyl epl witqn jk jxevl .qpkzn `ed

.0 ≤

P∞

n=1

ln(n) n2 xehd ik gikedl mivex √ ep` :x = 1 ok lre ,ln(n) ≤ n ik miiwzn

•

–

ln(n) 1 ≤ 3/2 n n

.d`eydd ogan i"tr ,qpkzn eply xehd -qpkzn .qpkzn ok lre ,hlgda qpkzn

P∞

1 n=1 n3/2 xehdy oeeikn

n

P∞

n=1

(−1) ln(n) xehd ,mcewd sirqd itl n2

x = −1 –

.[−1, 1] `ed zewfgd xeh ly zeqpkzdd megz -mekiql

4 dl`y Z

3 ln(3)

f (x)dx

.

lxbhpi`d jxr z` eayge ,(1, ∞) megza dtivx

2 ln(2)

fn (x)

f (x) =

P∞

n=1

ne−

nx 2

zeivwpetd ik ze`xdl epl witqi ,(1, ∞)-a dtivx dpezpd divwpetd ik ze`xdl ick .fn (x)

1

divwpetd ik e`xd .1

= ne−

nx 2

onqp

•

.megza y"na zeqpkzne ,zetivx olek od ogana ynzydl dvxp y"na zeqpkzd ze`xdl ick .(zeixhpnl` olek) ziciin `id fn (x) zeivwpetd ly ozetivx P∞ ik miiwzn n ∈ N lkly jk , n=1 an qpkzn ixtqn xeh `vnl epilr ,xnelk ,q`xhyxiie

.∀x ∈ (1, ∞), 0 < ne− miiwzn

x>1⇒−

‫גוריון‬-‫ בן‬,‫אגודת הסטודנטים‬

nx 2

< an

x ∈ (1∞)

lkl ik `ceel lw .an

= ne−n/2

z` gwip

n nx nx n < − ⇒ ne− 2 < ne− 2 x 2

2

‫מאגר הסיכומים‬

3 P∞ n/2 ik ze`xdl xzep dxcqd ik al miyp zeyrl zpn-lr .qpkzn n=1 ne x/2 :xiyi aeyig i"r heyt e` ,g(x) = xe divwpetd zxiwg i"r z`f ze`xl ozip) n ≥ 2 xear ,zipehepen

oke ,0-l zt`eye ziaeig `id

an

n + 1 −n−1+n an+1 n+1 √ < 1 ⇐⇒ = e 2 < e an n n .ilxbhpi`d d`eeydd ogan z` oezpd xehd lr lirtdl ozip -ok lr .(n

Z

≥2

miiwzn ok`y dn

 u=x du = dx v = −2xe−x/2 dv = e−x/2 dx ∞ ∞ Z ∞ = −2xe−x/2 +2 e−x/2 dx = −2e−x/2 (x + 2) = 6e−1/2 < ∞

xex/2 dx =

1∞



1

x=1

x=1

.dtivx .(1, ∞)-a lkend xebq rhw-zz lk lr y"na qpkzn `ed ,(1, ∞) lr mb y"na qpkzn ,xai` -xai` divxbhpi` rval ozip df megza ,ok lr .[2 ln(2), 3 ln(3)]

⊆ (1, ∞)

f

P∞

divwpetd okle

n=1 fn (x)

xehdy oeeikn

•

lr y"na qpkzn xehd ,hxta ik miiwzne

Z

3 ln(3)

Z

∞ X

3 ln(3)

f (x)dx = 2 ln(2)

2 ln(2)

=

∞ X

n·

n=1

= −2

! ne

− nx 2

dx =

∞ X

n

=0

divwpetl

dx

 n·2 ln(2) −2  −n·3 ln(3) 2 − e− 2 e n

∞  X

!

3

− 23 n

3

−n

−2



3− 2

=2 1−

3

1 − 3− 2

n=1

.f (x)

e

− nx 2

2 ln(2)

n=1

n=1

!

3 ln(3)

Z

[0, 2π]



rhwa y"na zqpkzn .

fn (x) = 1 − cos

x n


∞ n=1

ik miiwzn

.∀x ∈ [0, 2π],



lkl - epayigy oeieeyd i` itl .

2π n0

2

lkl ik al miyp

π x ∈ [0, ] ⇒ cos ≥0 n 2 n

 x  (1 − cos(x/n)) (1 + cos(x/n)) cos(x/n)≥0 sin2 (x/n) . 1 − cos ≤ = n 1 + cos(x/n) 1 n > n0

n≥4

x

ik miiwzn

-e

dxcqd ik egiked .2

< -y

jke

n0 ≥ 4-y

jk

x ∈ [0, 2π]-e n ≥ 4

|sin(x)| 0 `di ,zrk x ∈ [0, 2π]

ik miiwzn

 x   2π 2  2π 2 <