Lecture 9 - Excellent notes for chapter 9 , calc 2 series. PDF

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Excellent notes for chapter 9 , calc 2 series....

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• • • •

∞ X n=0

an

an (x − c)n , c

R

|x − c| < R

R

1 f (x) = 1−x

•

|x| < 1

1 = 1 + x + x2 + x3 + x4 + · · · 1−x c=1

r=x

f

c f

f (x) f (x) =

∞ X f (n) (c) n=0

n!

(x − c)n = f (c) +

c f ′ (a) f ′′(a) (x − a) + (x − a)2 + · · · . 1! 2!

c

c

an =

f (n) (c) n!

TN (x)

TN (x) = TN (x)

N X f (N ) (c) f (n) (c) f ′ (c) (x − c) + · · · + (x − a)N . (x − c)n = f (c) + 1! N ! n! n=0

N c = 0 f f f f (x)

TN (x)

N

f (x)

c=

•

f (x) =

π 2

∞ X f (n) (c) n=0

f (x) = sin x f ′ (x) = cos x f ′′(x) = − sin x f ′′′(x) = − cos x f (4)(x) = sin x ···

sin x =

∞ X sin(n) (x) n=0

n!

f (x) = sin(x)

n!

(x − c)n .

f (c) = f (π/2) = sin(π/2) = 1 f ′ (c) = f ′ (π/2) = cos(π/2) = 0 f ′′ (c) = f ′′(π/2) = − sin(π/2) = −1 f ′ (c) = f ′ (π/2) = − cos(π/2) = 0 f (4)(c) = f (4)(π/2) = sin(π/2) = 1 ···

(x − π2 )n

sin′′( π2 ) sin′ ( π2 ) π + (x − 2π)2 (x − 2 ) + = 2! 1! 0 1 0 −1 0 (x − π2 )2 + (x − π2 )3 + (x − π2 )4 + (x − π2 )5 + · · · = 1 + (x − π2 ) + 5! 4! 3! 2! 1! ∞ n X (−1) (x − 2π )2n = (2n)! n=0 sin( 2π )

f (x) = ex

• c=0

n f (n) (x) = ex

=⇒

f (n) (c) = f (n) (0) = e0 = 1.

f (x) ex =

∞ X f (n) (c) n=0 ∞

=

n!

(x − c)n

X 1 (x)n n! n=0

= 1+x+

x4 x2 x3 + + + ··· 3! 2! 4!

f (x) = x2 ex

•

ex ex =

∞ X xn . n! n=0

x2 x2 ex = x2

∞ X xn

n!   x2 x3 x4 2 + ··· + + = x 1+x+ 4! 3! 2! x6 x4 x5 + + ··· = x2 + x3 + + 3! 2! 4! ∞ X xn = (n − 2)! n=0

n=2

∞ X x2n+1 x3 x5 x7 + ··· (−1)n • sin x = − + = x− 7! 5! 3! (2n + 1)! n=0

• cos x =

∞ X

(−1)n

n=0

x2 x4 x6 x2n − + ··· + = 1− 4! 2! (2n)! 6!

∞

X 1 • = xn = 1 + x + x2 + x3 + x4 + · · · 1−x n=0

x

• e =

∞ X xn n=0

n!

= 1+x+

x3 x2 + + ··· 2! 3!

• arctan(x) =

∞ X

• ln(1 + x) =

∞ X xn+1 x4 x2 x3 (−1)n − + = x− + ··· 3 2 n+1 4 n=0

n=0

(−1)n

x3 x5 x7 x2n+1 − + = x− + ··· 5 3 2n + 1 7

• x≈0 sin x ≈ x,

ex ≈ 1 + x.

tan x ≈ x,

sin x →1 x

x → 0. R

sin(2.704)

∗ ∗

2

ex dx

π eiπ + 1 = 0

f TN (x) N X f (N ) (c) f (n) (c) f ′ (c) (x − c) + · · · + (x − a)N . TN (x) = (x − c)n = f (c) + 1! N ! n! n=0

RN (x)

RN (x) =

X f (n) (c) f (N +1)(c) f (N +2)(c) (x − c)n = (x − a)(N +1) + (x − a)(N +2) + · · · . n! (N + 1)! (N + 2)! n=N +1 N

f (x) = TN (x) + RN (x). M |f (x) − TN (x)| = |RN (x)| ≤

|f

(N +1)

(x)| ≤ M

M |x − a|N +1 . (N + 1)!

f (N +1)(x) |x − a| ≤ d d

sin x sin(2) c=

π 2

sin x =

∞ X (−1)n n=0

T6 (x) = 1 −

(2n)!

(x − π2 )2n .

1 1 1 (x − π2 )2 + (x − π2 )4 − (x − π2 )6 . 6! 4! 2!

x=2 sin(2) ≈ T6 (2) = 1 −

1 1 1 (2 − 2π )2 + (2 − π2 )4 − (2 − 2π )6 = 0.9092973983 . . . 6! 4! 2!

| sin(2) − T6 (2)| = |R6 (2)| ≤ M | − cos x| ≤ 0.4 |R6 (2)| ≤

M |2 − π2 |7 . 7!

sin(7)(x) = − cos x [ π2 , 2]

M |2 − π2 |7 = 2.2154 × 10−7 . 7!

sin(2) = 0.9092974268 . . .

sin(x) sin x =

∞ X

(−1)n

n=0

T6 (x) = x −

x2n+1 . (2n + 1)!

x3 x5 + , 5! 3!

M = 0.4

sin(2) ≈ T6 (2) ≈ 2 −

π 2

sin(2) π 2

T6 (x)

T6 (x)

π 2

23 25 + = 0.93. 3! 5!

sin(x) T6 (x) R1 0

ex = x2

2

ex dx

∞ X xn . n! n=0

x 2

ex =

Z Z

x2

e

1

x2

e

∞ ∞ X (x2 )n X x2n = . n! n! n=0 n=0

Z X ∞ ∞ X x2n x2n+1 = + C. dx = n! (2n + 1)n! n=0 n=0

dx =

0

∞ hX n=0

∞ X

x2n+1 i1 (2n + 1)n! 0

∞

X 02n+1 12n+1 = − (2n + 1)n! n=0 (2n + 1)n! n=0 = 1+

1 1 1 + ··· + + 42 3 10

≈ 1.46

π π π ∞ X 1 π2 = . n2 6 n=1

π π arctan(1) = . 4 arctan(x) arctan(x) = x −

x7 x3 x5 − + + ··· 5 3 7

  1 1 1 π = 4 arctan(1) = 4 1 − + − + · · · . 3 5 7

1 arctan(1) = 4 arctan(51) − arctan(239 ) 1 5

1 )). π = 4(4 arctan(51) − arctan( 239

=⇒ 1 239

c=0

i=

√

eiπ + 1 = 0 −1,

i2 = −1,

i3 = −i, ex =

eix = 1 +

i4 = i3 ·i = (−i)i = −(−1) = 1,

∞ X xn n! n=0

i5 = i,

∞ X (ix)n . e = n! n=0 ix

=⇒

x6 ix7 −ix3 x4 ix5 ix x2 + ··· − − + + − − 7! 6! 5! 4! 3! 2! 1! ∞

X x2n x2 x4 x6 − + = cos x. + ··· = (−1)n 1− 4! 2! (2n)! 6! n=0 ∞ X ix7 x2n+1 ix ix3 ix5 − + − (−1)n + ··· = i 5! 3! 1! (2n + 1)! 7! n=0

!

= i sin x.

eix = cos x + i sin x. x=π eiπ = cos π + i sin π = −1 + 0

=⇒

eiπ + 1 = 0.

i6 = −1, . . ....