Can you verify if my answer is correct for this problem?
Answer: sqrt(5)/sqrt(5)-1
Hello, l flipboi l
Can you verify if my answer is correct for this problem?
. . $\displaystyle \sum^{\infty}_{n=0} \frac{1}{(\sqrt{5})^n}$
$\displaystyle \text{Answer: }\:\frac{\sqrt{5}}{\sqrt{5}-1}$ . Yes!
$\displaystyle \text{Infinite geometric series with first term }a = 1\,\text{ and common ratio }r = \tfrac{1}{\sqrt{5}}$
. . $\displaystyle \text{Sum} \:=\:\frac{a}{1-r} \;=\;\frac{1}{1-\frac{1}{\sqrt{5}}} \;=\;\frac{\sqrt{5}}{\sqrt{5}-1} $
$\displaystyle S=\frac{1}{\sqrt{5}^0}+\frac{1}{\sqrt{5}^1}+...... .$
$\displaystyle S-1=\frac{1}{\sqrt{5}^1}+\frac{1}{\sqrt{5}^2}+.....$
$\displaystyle \sqrt{5}(S-1)=1+\frac{1}{\sqrt{5}^1}+....$
$\displaystyle \sqrt{5}S-\sqrt{5}=S$
$\displaystyle S\left(\sqrt{5}-1\right)=\sqrt{5}$
$\displaystyle S=\frac{\sqrt{5}}{\sqrt{5}-1}$