Does a series have to start from 1 to be able to apply convergence tests? In the ratio test, for example, we know that the test applies for a sum from n=1 to n=infinity.
Can I directly apply this test to sums from n=x to n=infinity (for example, a sum from n=0 -> infinity)?


The behavior of finitely many terms never affects convergence of sums like this, so the fact that the index starts at 0 rather than 1 is irrelevant in applying the test. 

