The Alternating Series Test Examples

Can we use the AST to conclude that the series

converges?

No, because this is not an alternating series. Since 2n is always even, the quantity

(-1)²ⁿ

is always positive. We can't use the AST here.

Can we use the AST to conclude that the series

Σ(-1)ⁿn

converges?

No. While this is an alternating series, the other conditions aren't satisfied. We have

|a_n| = n < n + 1 = |a_{n + 1}|

which is the opposite of what we would need to use the AST. Also, the terms don't converge to zero. We can't use the AST here. In fact, since the terms don't converge to zero, the divergence test tells us the series diverges.

Consider the alternating series

Find an upper bound for the error in the 3rd partial sum.

We can use the AST to show that the series does converge. Let L be the value it converges to. The sequence of partial sums jumps the line at height L a few times. The 3rd partial sum S₃ happens to land above the line.

On the 4th jump, the partial sum jumps by

and overshoots the line at height L:

Since we overshot the line, the line must be closer than  to the 3rd partial sum.

In symbols,

This means the approximation S₃ is within 0.25 of the real sum of the series.