Determine whether the series is convergent or divergent

Question

$$\sum_{n=0}^{\infty} \frac{1}{x^n}$$ $$ \sum_{n=0}^{\infty} \frac{1}{x^n}= \sum_{n=0}^{\infty} \: \left( \frac{1}{x} \right)^n = \left(\frac{1}{x}\right)^n=\frac{a}{1-r} = \frac{x}{x-1} \: when \: |x|>1 $$

It is convergent.

The above is the textbook's solution...

Here's my work :

$$ lim_{n \to \infty}\frac{1}{x^n}=\frac{1}{\infty}=0 \:\:when\;x\geq1 $$ $$It's\;\; divergent \;\;by\;\;the\;\;n^{th} test$$ Where's my mistake ?

Answer 1

I don't know what you think the $n^{\text{th}}$ test is, but taking the limit of the terms should yield only two conclusions

• If the limit is not $0$, then the series is divergent.
• If the limit is $0$, the test is inconclusive (we know nothing about the series' behavior). Another test is needed.