Answer to Question #91365 in Real Analysis for Sajid

Question

Q. Choose the correct answer.
Q. Which of the following is the sum of infinite series ∑_(n=0)^∞▒〖(-1)〗^n 3^n r^n?
a. 1/(1+3r) for -1/3<r<1/3
b. . 1/(1-3r) for -1/3<r<1/3
. 3/(1-3r) for -1/3<r<1/3
d. ∞

Answer 1

$$1/(1-x)=1+x+x^2+...+x^n=\sum_{n=0} ^\infin x^n, |x|<1$$

- the geometric series.

Then

$$1/(1+x)=1/(1-(-x))=$$

$$=1-x+x^2+...+(-1)^nx^n=\sum_{n=0} ^\infin (-1)^nx^n, |x|<1$$

If

$$x=3r$$

then

$$\sum_{n=0}^\infin (-1)^n 3^nr^n=1/(1+3r)$$

for

$$|3r|<1 \Rightarrow - 1/3<r<1/3.$$