1. ## geometric series

Problem 1

n=0 to infinity (-1/2)^n (x-3)^n

How would you find the value of x for which the given geometric series converges and how would you find the sum of the series(as a function of x) for the value of x?

Problem 2

How would you find the values of b for which

1+ e^b+ e^2b+ e^3b+.....=9

2. ## GP

Hello TAG16
Originally Posted by TAG16
Problem 1

n=0 to infinity (-1/2)^n (x-3)^n

How would you find the value of x for which the given geometric series converges and how would you find the sum of the series(as a function of x) for the value of x?
$\displaystyle (-\tfrac{1}{2})^n(x-3)^n = (-\tfrac{1}{2}(x-3))^n =\left(\frac{3-x}{2}\right)^n$

So the series converges if $\displaystyle -1 < \frac{3-x}{2}<1$

$\displaystyle \Rightarrow -2<3-x<2$

$\displaystyle \Rightarrow -5<-x<-1$

$\displaystyle \Rightarrow 5>x>1$

The sum to infinity $\displaystyle S= \frac{a}{1-r}$, where $\displaystyle a = 1$ and $\displaystyle r = \frac{3-x}{2}$

$\displaystyle \Rightarrow S =$ ... you can finish it now.

Problem 2

How would you find the values of b for which

1+ e^b+ e^2b+ e^3b+.....=9
This is a GP where $\displaystyle a = 1$ and $\displaystyle r = e^b$. So the sum to infinity $\displaystyle S= \frac{a}{1-r} = \frac{1}{1 - e^b} = 9$

Solve this equation for $\displaystyle b$. (The answer will involve $\displaystyle \text{ln}$.)