Help me solve these multiple questions on taylor expansion

Please show the solution to these:
Find the Taylor expansion at (0,0) of the following functions.

1. 1/(1-X1-X2)

2.e^(X1 + X2)

3. sin(X1 + X2)

The quick way to do these is to use the usual Maclaurin expansions for one variable.

1) 1/(1 - x₁ - x₂)
= 1/(1 - (x₁ + x₂))
= Σ(n = 0 to ∞) (x₁ + x₂)^n, using the usual geometric series
= Σ(n = 0 to ∞) Σ(k = 0 to n) C(n, k) (x₁)^(n-k) (x₂)^k, via binomial theorem
= 1 + (x₁ + x₂) + (x₁² + 2x₁x₂ + x₂²) + (x₁³ + 3x₁²x₂ + 3x₁x₂² + x₂³) + ...

2) e^(x₁ + x₂)
= Σ(n = 0 to ∞) (x₁ + x₂)^n / n!, using the usual exponential series
= Σ(n = 0 to ∞) (1/n!) * Σ(k = 0 to n) C(n, k) (x₁)^(n-k) (x₂)^k, via binomial theorem
= 1 + (x₁ + x₂) + (1/2!)(x₁² + 2x₁x₂ + x₂²) + (1/3!)(x₁³ + 3x₁²x₂ + 3x₁x₂² + x₂³) + ...

3) sin(x₁ + x₂)
= Σ(n = 0 to ∞) (-1)^n (x₁ + x₂)^(2n+1) / (2n+1)!, using the usual sine series
= (x₁ + x₂) - (1/3!)(x₁³ + 3x₁²x₂ + 3x₁x₂² + x₂³) + ...

I hope this helps!