Find a power series for the function: f(x)=1/(2x-5)

Answer 1

Write 1/(2x-5) in the form a/(1-r), the sum of a geometric series:

1 / (2x-5) = -1 / (5(1 - (2x/5)) = (-1/5) / (1 - (2x/5)

This is the sum of the geometric series ∑_{(n=0 to ∞)} arⁿ, where a = -1/5 and r = 2x/5

= -∑_{(n=0 to ∞)} (1/5)(2x/5)ⁿ

The series converges only when -1 < r < 1.

That is -1 < 2x/5 < 1

So, -5/2 < x < 5/2

Answer 2

this is a way to proceed ;

1/(2x-5) same as (-1)(1(/5-2x))

using synthetic division

correction

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

update: it can be seen in graph of y=1/(2x-5)

and y=(-1)(1/5+2x/5+4x²/5+8x³/5+16x⁴/5)

that the two curves are similar between -5/2<x<5/2