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 <