Look at the arithmetic series
ai = a1 + (i – 1)d
and d is negative. Does this series converge or diverge?
If a1 and d are both negative, every term will be negative and at most a1. In symbols,
ai ≤ a1 < 0
for all i. If we sum n such terms, we see that the nth partial sum is
Sn ≤ n(ai).
As n approaches infinity, the partial sums Sn keep getting more and more negative, so
This means the series diverges.
Just like the previous exercise, it doesn't matter if a1 is positive or negative. If we keep adding a negative value to a1, eventually we'll get a negative term an. From what we just said, the series
must diverge. Since changing the starting limit of summation doesn't change whether the series diverges,
has to diverge also.
A constant series diverges unless the constant is 0. By comparing arithmetic series to a constant series, we showed that arithmetic series have to diverge unless both a1 and d are 0.