Introduction to Series - At A Glance:
If we care about the value of a convergent series we need to specify the starting index. If we only care whether or not a series has a sum (that is, whether the series converges or diverges), we can ignore the limits of summation and write the series as
That's because the starting limit of summation doesn't affect whether a series converges or diverges.
If it sounds like we are waving our hands and pulling a lion out of a baseball cap, it might be worth looking at convergence of sequences. We know the tail of a sequence tells us about convergence, and the sequence of terms must go to zero for the series to converge. This means that we can ignore the head of the sequence, which happens to be the first terms of the series. The starting limit doesn't matter.
This means, for an infinite series with general term an either converges or diverges, regardless of which term we start at, if
converges, so does
and vice versa.
This fact saves us effort and headaches. It saves us effort because instead of having to write
we can be lazy (or efficient!) and write
It saves us headaches because we don't have to worry what we write for the starting limit of summation.
Suppose we're asked to show that
converges, but it's easier to show that
converges. We can wave our hands, do the one that's easier, and say, "since the starting limit of summation doesn't affect whether the series converges or diverges,
Be Careful: The choice of starting limit of summation doesn't affect whether a series converges or diverges. However, it will affect the final sum of the series. If you're asked to find a sum (not just to say whether the sum exists), you do have to make sure your starting limit is correct.