Series de Potencias 2015

8/18/2019 Series de Potencias 2015

1/16


2/16

y = f (x)

pn(x)

pn(x) = a0 + a1x + a2x2 + ... + anx

n.

n

x

n.

A(0, f (0))

f (x)

(n + 1)

x = 0

y = p0(x)

A a0 A

a0 = f (0) x = 0.

y = a0 + a1x

A

A

a0 = f (0) y a1 = f (0).


3/16

n y = a0 + a1x + a2x2 + ... + anx

n,

A

A

1 2 3 n

y = pn(x) n

x = 0

f (x)

x = 0

f (x)

pn(0) = f (0), p

n(0) = f (0), pn(0) = f

(0) ... pnn(0) = f n(0)

f (0) = a0, f (0)

1! = a1,

f (0)

2! = a2 ...

f n(0)

n! = an

f

n

x0

p(x) = f (x0) + f (x0)(x − x0)

1! +

f (x0)(x − x0)2

2! + ... +

f n(x0)(x − x0)n

n!

f

x0

x0 = 0

p(x) = f (0) + f (0)x

1! +

f (0)x2

2! + ... +

f n(0)xn

n!

f

f (x) = sen(x)

sen(x)

cos(x), −sen(x), −cos(x), .... x = 0

1, 0, −1, 0, .... x x − x3

3!

f (x) = sen(x) x = 0


4/16

f (x) = sen(x)

sen(x) ∼ x

f (x) = log(x + 1)

f (x) = log(x + 1)

x = 0

f (x)

n + 1

I

x0 Rn(x)

f (x) P n(x) Rn(x) = f (x) − P n(x)

f (x) = P n(x) + Rn(x)

f (x) = f (x0) + f (x0)

1! (x − x0) +

f (x0)

2! (x − x0)

2 + ... + f n(x0)

n! (x − x0)

n + Rn.

Rn(x)

Rn(x) = f n+1(ξ )

(n + 1)!(x − x0)

n+1 ξ x0 y x

f (x) = f (x0) + f (x0)

1! (x − x0) +

f (x0)

2! (x − x0)

2 + ... + f n(x0)

n! (x − x0)

n + f n+1(ξ )

(n + 1)!(x − x0)

n+1.

f (x)

x0 = 0

f (x) = f (0) + f (0)

1! x +

f (0)

2! x2 + ... +

f n(0)

n! xn +

f n+1(ξ )

(n + 1)!xn+1.

ξ

x0 x


5/16

Rn(x) I

Rn(x) = f (x) − f (x0) + f (x0)

1! (x − x0) + f (x0)

2! (x − x0)2 + ... + f n(x0)

n! (x − x0)n

∗

x

x0 g(t) I

g(t) = f (x) − f (t) − f (t)

1! (x − t) −

f (t)

2! (x − t)2 − ... −

f n(t)

n! (x − t)n − Rn(x)

(x − t)n+1

(x − x0)n+1

g(x) = 0

g(x0) = [f (x)−f (x0)−f (x0)

1! (x − x0)−

f (x0)

2! (x−x0)

2−...−f n(x0)

n! (x−x0)]−Rn(x)

(x − x0)n+1

(x − x0)n+1 = 0

∗ Rn(x)

x

x0 I g(x) = 0 g(x0) = 0

ξ

x

x0 g(ξ ) = 0

g(t)

g(t) = 0 − f (t) − [−f (t)

1! +

f (t)

1! (x − t) − 2

f (t)

2! (x − t) +

f (x0)

2! (x − x0)

2 − ...

−nf n(t)

n! (x − t)n−1 +

f n+1(t)

n! (x − t)n] + (n + 1)Rn(x)

(x − t)n

(x − x0)n+1

g(t) = −f n+1(t)

n! (x − t)n + (n + 1)Rn(x)

(x − t)n

(x − x0)n+1

g(ξ ) = 0

g(ξ ) = −f n+1(ξ )

n!

+ Rn(x) n + 1

(x − x0)n+1

= 0

Rn(x) = f n+1(ξ )

(n + 1)!(x − x0)

n+1.

f (x) = ex

e

ex

= 1 + x + x2

2! + ...


6/16

P 5(x) = 1 + x + x2

2! +

x3

3! +

x4

4! +

x5

5!

ex

e ∼ 1 + 1 + 1

2 +

1

6 +

1

20+

1

120 + R5(1)

e ∼ 2, 716 + R5(1)

R5(1) = eξ

(5 + 1)!(1)5+1

0 < ξ


7/16

∞n=0

anxn

0

∞n=0

anxn

x

x = 0.

x

(−R, R)

|x| > R.

∞n=0

xn = 1 + x + x2 + ... + xn + ...

(−1, 1)

x ∞n=0

xn = 1

1 − x

(−1, 1) ∞n=0

xn

1

1 − x.

R

∞n=0

anxn

x ∈ (−R, R)

x |x| > R.

(−R, R)

x

x = 0


8/16

∞n=0

anxn

L = ĺımn→∞

an+1an 0 ≤ L ≤ ∞

L = ∞ x = 0

L = 0 x

0 < L < ∞ R = 1

L |x| < R

−R < x < R |x| > R

∞n=0

xn

n! x

x = 0 1

x = 0 L = ĺımn→∞

1

(n + 1)!1

n!

= lı́m

n→∞1

n + 1 = 0

L = 0

R = ∞ x

x = 0

∞n=0

n! xn x = 0 L = ĺımn→∞

(n + 1)!n! = lı́mn→∞(n + 1) = ∞

x = 0

∞n=0

2nxn

n

−

1

2, 1

2

(x − x0)

∞n=0

an(x − x0)n = a0 + a1(x − x0) + a2(x − x0)

2 + ... + an(x − x0)n + ...

x0

x0 = 0 x


9/16

x

x − x0

∞n=0

an(x − x0)n

|x − x0| < R (x0 − R, x0 + R)

|x − x0| > R

∞n=0

(x − 1)n

n

L = ĺımn→∞

1

n + 11

n

= 1

|x − 1| < 1 x ∈ (0, 2) |x − 1| > 1

x = 0, ∞n=0

(−1)n

n

x = 2

∞n=0

(1)n

n

∞

n=0xn

n [0, 2)

(x − x0)

x0.

f (x)

n=0

an(x − x0)n

(x0 − R , x0 + R) R > 0

(x0 − R , x0 + R) f (x)

(x0 − R , x0 + R)


10/16

f (x) =∞n=0

an(x − x0)n

R

f |x − x0| < R

f

f (x) =∞n=1

ann(x − x0)n−1 = a1 + 2a2(x − x0) + ... + ann(x − x0)

n−1 + ... |x − x0| < R

f

ˆ f (x)dx =

∞n=0

´ an(x − x0)

ndx =∞n=0

an(x − x0)

n+1

n + 1 + C

|x − x0| < R

f (x) =∞n=0

xnn!

x

x0 = 0 R = ∞

f (x) = f (x)

x

f (x) = ex

L = 0 x.

x.

f (x) = d

dx

1 + x +

x2

2! +

x3

3! ...

= 0 + 1 +

2x

2! +

3x2

3! + ... = 1 + x +

x2

2! + ... = f (x)

f (x) = f (x)

f (x) = C ex

f (0) = 1

C = 1 f (x) = ex

∞n=0

xn 1

1 − x

|x| < 1 1

(1 − x)2

1

(1 − x)2

f (x) = 1

1 − x

f (x) = 1

(1 − x)

2 = 1 + 2x + 3x2 + 4x3 + ... + nxn−1 + ... , |x| < 1

∞n=0

xn+1

n + 1 = −ln(1−x) −1 < x


11/16

ˆ x0

1

1 − u du = −ln(1 − x)

−ln(1 − x) =

ˆ x0

1

1 − u du =

∞n=0

xn+1

n + 1

ln(x + 1) =∞n=0

(−1)n x

n+1

n + 1 (−1, 1)

f

x0 f

x − x0 x0

ε > 0 / f (x) =∞n=0

an(x − x0)n ∀x ∈ (x0 − ε , x + ε)

P (x) = a0 + a1x + a2x2 + ... + anxn

f (x) =

1

1 − x

x = 0

f (x) =

1

1 − x =

∞

n=0x

n

(3)

f

∞n=0an(x − x0)

n

f

I

f (x) =∞n=0

an(x − x0)n ∀x ∈ I .

f


12/16

f

f (x) =

∞n=0

an(x − x0)n en I = (x0 − R, x0 + R)

an = f n(x0)

n! para n = 0, 1, 2,...

f (x) = a0 + a1(x − x0) + a2(x − x0)2 + ... + an(x − x0)

n + ... ; f (x0) = a0

f (x) = 1· a1 + 2a2(x − x0) + ... + nan(x − x0)n−1 + ... ; f (x0) = 1· a1

f (x) = 1· 2· a2 + ... + n(n − 1)an(x − x0)n−2 + ... ; f (x0) = 1· 2.a2

f (x) = 1· 2· 3· a3 + ... + n(n − 1)(n − 2)an(x − x0)n−3 + ... ; f (x0) = 1· 2· 3· a3

an = f (n)(x0)

n!

f (x) = f (x0) + f (x0)

1! (x − x0) +

f (x0)

2! (x − x0)

2 + ... + f n(x0)

n! (x − x0)

n + ...

f

f


13/16

f

I

x0 ∈ R

f

x0

∞n=0

f (n)(x0)

n! (x − x0)

n = f (x0) + f (x0)(x − x0)

1! +

f (x0)(x − x0)2

2! + ... +

f (n)(x0)(x − x0)n

n! + ...

f

x0 = 0

∞n=0

f (n)(0)n!

xn = f (0) + f

(0)1!

x + f (0)2!

x2 + ... + f (n)(0)

n! xn + ...

Rn(x) = f (x) − P n(x), P n(x)

n ĺımn→∞

Rn(x) =

0 ∀ x ∈ I f (x) P n(x)

Rn(x).

f

(x0 − R, x0 + R

f (x) = f (x0) + f (x0)

1! (x − x0) +

f (x0)

2! (x − x0)

2 + ... + f n(x0)

n! (x − x0)

n + ...

f

lı́mn→∞

Rn(x) = 0

Rn(x) = f n+1(ξ )

(n + 1)!(x − x0)

n+1 ξ (x0 − R, x0 + R

ĺımn→∞

Rn(x) = 0

f (x) = P n(x) + Rn(x) n → ∞,

f (x)

ξ

ξ

f

|Rn(x)|


14/16

f −R < x − x0 < R

f

x = x0

f

x0

f

g

f −R < x − x0 < R

f

x = 0

f (n)(0) = 0 , ∀n

0 + 0

1x +

0

2!x2 +

0

3!x3 + ... = 0

f (x) = 0 ∀x f (x) x = 0

f (x) =

e− 1

x2 x = 0

0 x = 0

R − {0}

x = 0

x = 0

x = 0

Rn(x) 0 n → ∞.

sen(x) = x − x3

3! +

x5

5! + ... + (−1)

n x2n+1

(2n + 1)! + ... =

∞n=0

(−1)n

(2n + 1)!x2n+1

x

cos(x) = 1 − x2

2! +

x4

4! + ... + (−1)n

x2n

(2n)! + ... =

∞n=0

(−1)n

(2n)! x2n x

ex = 1 + x + x2

2! +

x3

3! +

x4

4! + ... +

xn

n! + ... =

∞n=0

xn

n!

x

e =∞n=0

1

n!


15/16

f (x) = 1

1 − x

∞n=0

xn = 1 + x + x2 + ... + xn + ..., ∀x ∈

(−1, 1) .

f (x) = 1(1 − x)2

f (x) = ln(x + 1)

∞n=0

(−1)n xn+1

n+1 , ∀x ∈ (−1, 1)

ln(1 − x) −x x

ln(x + 1)

ex = 1 + x + x2

2! +

x3

3! +

x4

4! + ... +

xn

n! + ... =

∞n=0

xn

n! x

lı́mn→∞

xn

n! = 0

f (x) = sen(x)

sen(x) = x − x3

3! + x5

5! + ... + (−1)n

x2n+1

(2n + 1)! + ... =

∞n=0

(−

1)n

(2n + 1)! x2n+1 x

x

ĺımn→∞

Rn(x) = ĺımn→∞

f n+1(ξ )

(n + 1)!(x − x0)

n+1

| f n+1(x) |=| cos(x) | | f n+1(x) |=| sen(x) | | Rn(x) |≤ | x |n+1

(n + 1)!

ĺımn→∞

xn

n! = 0

ĺımn→∞

Rn(x) = 0

f

,

e−x2

e

x

=

∞n=0

xn

n!


16/16

´ e−x

2

dx =∞n=0

(−1)n x2n+1

(2n+1)n! + C

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