Look at the arithmetic series

where

*a*_{i} = *a*_{1} + (*i* – 1)*d*

and *d* is negative. Does this series converge or diverge?

Answer

If *a*_{1} and *d* are both negative, every term will be negative and at most *a*_{1}. In symbols,

*a*_{i} ≤ *a*_{1} < 0

for all *i*. If we sum *n* such terms, we see that the *n*th partial sum is

*S*_{n} ≤ *n*(*a*_{i}).

As *n* approaches infinity, the partial sums *S*_{n} keep getting more and more negative, so

This means the series diverges.

Just like the previous exercise, it doesn't matter if *a*_{1} is positive or negative. If we keep adding a negative value to *a*_{1}, eventually we'll get a negative term *a*_{n}. 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 *a*_{1} and *d* are 0.