I am given the following series to determine if it is convergent. My problem is, I use 2 tests and once I get that the set is convergent, and the other time divergent.
The series is $$\sum_{n=0}^{\infty} \left(a^n-a^{n+1}\right)$$
The root test: $$\lim_{n\to\infty} \left(a^n-a^{n+1}\right)^{1/n}=\lim_{n\to\infty} \left(a-a^{{\frac{1}{n}}+1}\right)=0$$
So this is convergent
The ratio test: $$\lim_{n\to\infty} \frac{a^{n+1}-a^{n+2}}{a^n-a^{n+1}}=a$$
This is divergent?
In both cases n goes to infinity
What is wrong with this?
Thanks in advance!
The ratio test gives you $a$, so you could definitely conclude that the series converges if $|a|<1$. However, more generally notice that $$\sum_{n=1}^\infty a^n-a^{n+1} = (a-a^2)+(a^2-a^3)+(a^3-a^4)+\ldots $$ Now regroup quantities carefully and see if you can get some things to cancel.
Lastly, as a general rule $(x+y)^z \neq x^z+y^z$.