Question

Find the power series representation for the function f(x)=x/4−x.

(a) ∞∑n=0x^n+1/4^n+1

(b) ∞∑n=0x^n+14^n

(c) ∞∑n=0x^n4^n

(d) ∞∑n=0x^n+1

I had been asked this question by my school principal while I was bunking the class.

The question is from Discrete Probability in chapter Discrete Probability of Discrete Mathematics

Answer 1

Right option is (a) ∞∑n=0x^n+1/4^n+1

For explanation: So, again, we’ve got an x in the numerator.  f(x)=x*1/4−x. If there is a power series representation for g(x)=1/4−x, there will be a power series representation for f(x). Suppose, g(x)=1/4*1/1−x^4. To get a power series representation is to replace the x with x^4. Doing this gives, g(x)=1/4 ∞∑n=0 x^n/4^n (x^n/4 nprovided ∣x/4∣<1) ⇒ g(x) = 1/4 ∞∑n=0 x^n/4^n = ∞∑n=0 x^n/4^n+1. The interval of convergence for this series is, ∣x/4∣<1⇒1/4  |x|<1⇒|x|<4. Now, multiply g(x) by x and we have f(x)=x*1/4−x=x ⇒ ∞∑n=0 x^n/4^n+1 = ∞∑n=0x^n+1/4^n+1 and the interval of convergence will be |x|<4.