proof of if x^n=0 then 1/(1-x)=1+x+x^2+...+x^(n-1)

1. NILPOTENT AND
INVERTIBLE
Hanpen Robot

2. Today’s topic !
1 − 𝑥 −1
= 1 + 𝑥 +∙∙∙ +𝑥 𝑛−1
.
𝑖𝑓 𝑥 𝑛
= 0 𝑡ℎ𝑒𝑛

3. Let’s proof !

4. 𝑆 = 1 + 𝑥 +∙∙∙ +𝑥 𝑛−1
𝑥𝑆 = 𝑥 +∙∙∙ +𝑥 𝑛−1
+ 𝑥 𝑛

5. 𝑆 = 1 + 𝑥 +∙∙∙ +𝑥 𝑛−1
𝑥𝑆 = 𝑥 +∙∙∙ +𝑥 𝑛−1
+ 𝑥 𝑛

6. 𝑆 = 1 + 𝑥 +∙∙∙ +𝑥 𝑛−1
𝑥𝑆 = 𝑥 +∙∙∙ +𝑥 𝑛−1
+ 𝑥 𝑛
Remember 𝑥 𝑛
= 0

7. 𝑆 = 1 + 𝑥 +∙∙∙ +𝑥 𝑛−1
𝑥𝑆 = 𝑥 +∙∙∙ +𝑥 𝑛−1
+ 0
Remember 𝑥 𝑛
= 0

8. 𝑆 = 1 + 𝑥 +∙∙∙ +𝑥 𝑛−1
𝑥𝑆 = 𝑥 +∙∙∙ +𝑥 𝑛−1
∴ 1 − 𝑥 𝑆 = 1

9. 𝑆 = 1 + 𝑥 +∙∙∙ +𝑥 𝑛−1
𝑥𝑆 = 𝑥 +∙∙∙ +𝑥 𝑛−1
∴ 𝑆 = 1 − 𝑥 −1

10. 1 − 𝑥 −1
= 1 + 𝑥 +∙∙∙ +𝑥 𝑛−1