X+∞ k=0 (−1)k x 2 2k (2k

7.17 1) sin ^x₂

+ cos ^x₂

= X+∞

k=0

(−1)^k

x 2

^2k+1 (2k+ 1)! +

X+∞

k=0

(−1)^k

x 2

^2k (2k)!

=

+∞

X

k=0

(−1)^k

x 2

^2k+1

(2k+ 1)! + (−1)^k

x 2

^2k (2k)!

= X+∞

k=0

(−1)^k

x 2

^2k

(2k)! + (−1)^k

x 2

^2k+1 (2k+ 1)!

= X+∞

k=0

(−1)^k 1

2^2k(2k)!x^2k+ (−1)^k 1

2^2k+1(2k+ 1)!x^2k+1

= 1 + 1 2x−

1

2²2!x²− 1

2³3!x³+ 1

2⁴4!x⁴+ 1

2⁵5!x⁵+. . .

2) cos²(x) = 1 + cos(2x)

2 = ¹₂ +¹₂ cos(2x) = ¹₂ +¹₂ X+∞

k=0

(−1)^k(2x)^2k (2k)!

= ¹₂ + ¹₂ X+∞

k=0

(−1)^k 2^2kx^2k

(2k)! = ¹₂ + X+∞

k=0

(−1)^k1₂ · 2^2k (2k)!x^2k

= ¹₂ + X+∞

k=0

(−1)^k 2^2k−1 (2k)!x^2k

= ¹₂ + ¹₂

| {z }

=1

− 2

2!x²+ 2³ 4!x³−

2⁵

6! x⁶+. . .+ (−1)^k 2^2k−1

(2k)!x^2k+. . .

Analyse : développement en série d’une fonction Corrigé 7.17