2) find the taylor series for h(x) = ln(4+8x) about x = 0. include the interval

Question

2) find the taylor series for h(x) = ln(4+8x) about x = 0. include the interval of convergeance 3) use a series to evaluate or approximate. find the exact value of 1 + 2 + 4/2! + 8/3! +16/4! ... 4) solve for x: x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ... = 1.

Explanation / Answer

Start from the geometric series: 1/(1 - t) = S(n= 0 to 8) t^n Let t = -x/4: 1/(1 - (-x/4)) = S(n= 0 to 8) (-x/4)^n ==> 4/(4 + x) = S(n= 0 to 8) (-1/4)^n x^n ==> 1/(4 + x) = S(n= 0 to 8) (-1)^n (1/4)^(n+1) x^n. Integrate both sides from 0 to t: ln |t + 4| = S(n= 0 to 8) (-1)^n (1/4)^(n+1) t^(n+1)/(n+1). Finally, let t = x^2: ln(x^2 + 4) = S(n= 0 to 8) (-1)^n (1/4)^(n+1) x^(2n+2)/(n+1). I hope this helps!

Related Questions

Navigate

• Browse (All)
• Browse 2
• Subjects

---

---

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