1. Make use of the well-known fact that 1/1-x = infinity Sigma n=0 x^n for |x| <

Question

1. Make use of the well-known fact that 1/1-x = infinity Sigma n=0 x^n for |x| < 1 to do the following. (a) Find the power series representation of f(x) = ln(1 + x2). What is the radius of convergence? (b) Estimate Integral from 0 to 1/2 (1/1+x^4)dx correct within 0.0001. For this second question, I managed to answer a, b and c .. just need help on doing D and E. 2. Consider four points P (1, ?2, 2), Q(3, ?3, 0), R(?1, 0, 3), and S(0, ?1, 1) in R3. ?? ?? (a) (5 pts) Compute the angle between P Q and P R to the nearest tenth degrees. ?? ?? (b) (2 pts) Find the vector projection PR onto PQ ?? ?? (c) (2 pts) Express PQ

Explanation / Answer

Let t = -x: 1/(1 + x) = S(n=0 to 8) (-1)^n x^n, convergent for |x| < 1. Integrate both sides from 0 to x: ln(1 + x) = S(n=0 to 8) (-1)^n x^(n+1)/(n+1). This also has radius of convergence 1. (b) Multiply the result of (a) by x on both sides: x ln(1 + x) = S(n=0 to 8) (-1)^n x^(n+2)/(n+1). (c) Replace x with x^2 in the result from (a) should do the trick.

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.