Question

1. find the sum, if possible:

     1+3+1/3+1/9+1/27+.........

Answer 1

Geometric series

$$\displaystyle\sum_{i=0}^\infin ar^n=\dfrac{a}{1-r}, |r|<1$$

We have $a=1, r=\dfrac{1}{3}, |r|=\dfrac{1}{3}<1.$ Then

$$\displaystyle\sum_{i=0}^\infin (\dfrac{1}{3})^n=1+\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+...=\dfrac{1}{1-\dfrac{1}{3}}=\dfrac{3}{2}$$

Therefore

$$1+3+\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+...=3+\dfrac{3}{2}=$$

$$=\dfrac{9}{2}=4.5$$