We have discussed whether a series will converge, when for each . But what about when we let be anything?
What happens to ? When we expand it out, we see that it looks like . But by comparison for each . And we know the series converges while diverges. So what happens to ?
1. Proposition. Let
be a given series where . Then the series converges if
- for each ;
Proof: We have in the sequence of partial sums
But we also have
Conclusion: . ▮
Example 1. Consider the series
We see that for each , and
Thus the Alternating Series Test implies the alternating Harmonic series converges.
Example 2. Consider the series
why? Well, multiply both sides by and we get
Cross multiplication gives us
Subtracting from both sides gives us
So our series satisfies the first condition for the alternating series test.
We also see that
for “large ”. So we see
Thus our series satisfies the criteria for the alternating series test, which implies convergence.
2. Definitions. A series is “Absolutely Convergent” (or “Converges Absolutely”) if converges.
On the other hand, if is divergent, then we call the series “Conditionally Convergent” (or we say the series “Converges Conditionally”).
Example 3. We see that is conditionally convergent, since diverges.
Example 4. The series is absolutely convergent since the integral test tells us converges.
Question: let be a convergent series, and for each . Does the series converge?
Stop and think before continuing!
Proof: We see first that
So what? Well, we see that
Since converges, we see converges too. But by the comparison test, we see
converges. So what? We haven’t proven converges, have we? Consider the following trick
Therefore the series converges. ▮
Example 5. Consider the series
What to do? We know
So what? We know converges by the integral test. Then
converges by comparison. By the absolute convergence test, we know our series in Equation (20) converges.