Homework Template

Problem 1—Series approximation

Consider the function ƒ(x)=(log_)(1+x).

Evaluate the sum (∑_k=1^n)(k) and derive a closed form expression.

(a) Derive the Taylor expansion

We expand around x=0. Using the identity

1/(1+x)=(∑_n=0^∞)(−1)*x^n,|x|<1

and integrating term-by-term, we obtain

(log_)(1+x)=(∑_n=1^∞)(−1)(x^n)/n

(b) Approximation

Using the first four terms,

(log_)(1+x)≈x−(x^2)/2+(x^3)/3−(x^4)/4

(c) Error analysis

The remainder after truncation at order n satisfies

(R_n)(x)=O(x^(n+1))

Thus for small x, the approximation error is dominated by the next term in the series.

Problem 2—Linear system and conditioning

Consider

A*x=b,A=[[2,1],[1,3]]

(a) Solve the system

We compute

A^(−1)=1/5*[[3,−1],[−1,2]]

Thus

x=A^(-1)*b

(b) Sensitivity to perturbations

Let b→b+δ*b. Then

x→x+δ*x=A^(−1)*(b+δ*b)

so

δ*x=A^(−1)*δ*b

(c) Interpretation

The magnitude of A^(-1) controls how perturbations in b affect the solution. This motivates the definition of the condition number

k(A)=‖A‖⋅‖A^(-1)‖

Problem 3—Optimization

Consider the function

ƒ(x)=x^2+4*x+4

(a) Find the minimum

ƒ′*(x)=2*x+4

Setting ƒ(x)=0, we obtain

x=-2

(b) Second-order condition

ƒ″*(x)=2>0

Thus x=-2 is a minimum.

(c) Gradient descent step

With step size η ,

(x_t+1)=(x_t)−η*(2*(x_t)+4)

Comments

• In Problem 1, the expansion follows from integrating the geometric series.

• In Problem 2, conditioning plays a central role in numerical stability.

• In Problem 3, the quadratic structure allows closed-form optimization.