We are learning series and I really need help with this. I can't find any help with these, could someone please help? Thanks!

Question

Answer 1

First do partial fraction decomposition

f(x) =-5/(3x+1) - 5/(2x-1)

We want the form such that

g₁/(1-r) = ∑_n=1^∞ g₁r^n-1

for both fractions

-5/(1-(-3x)) +5/(1-2x)

f(x) = ∑-5(-3x)^n-1 + ∑5(2x)^n-1

n

1 -5 + 5 = 10

2 15x + 10x = 25x

3 -45x² + 20x² = -25x²

---etc---

Run out for n=1 to 5

f(x) ≈ 10 + 25x - 25x²+-......

Answer 2

f(x) = 25x/(1 + x - 6x²) = 25x/{1- 6[(x - 1/12)² - 1/144]} = 25x//[25/24 - 6(x - 1/12)²] = 24x/[1 - 144/25(x - /12)²]

This expression we can treat as sum of infinite geometric sequence with first term is 24 x and common ratio

is [12/5(x - 1/12)]² < 1; Ix - 1/12I < 5/12; -1/3 < x < 1/2