Since both and converge, we can break up the series:
If or diverged, we wouldn't be allowed to break up the series like that (and it wouldn't help us anyway).
Example 2
Find the sum of the series.
You may assume
Answer
Since converges and 5 is a constant, we're allowed to pull the constant out of the summation sign:
Example 3
Find the sum of the series.
You may assume
Answer
Remember that -bn is the same thing as (-1)bn. Since converges, we can pull the (-1) out of the summation sign:
Example 4
Find the sum of the series.
You may assume
Answer
This is a combination of parts (a) and (c). Since both and converge (this is necessary!), we can break up the sum:
Example 5
Find the sum of the series.
You may assume
Answer
Since Σ cn diverges, we aren't allowed to break up the sum. However, in this case, we don't have to. The quantity
cn – cn
is always zero, so
Adding up zero infinitely many times gives us zero.
Example 6
Let be a convergent series and be a divergent series. Does the series
converge or diverge? Does the answer depend on the particular choice of an and/or bn?
Hint
Suppose converges. What happens?
Answer
Following the hint, suppose
converges. Since we know converges also, we can say
The left-hand quantity is the difference of two convergent series, which is some finite number.
The right-hand quantity is
which diverges.
We're claiming a finite number is equal to the sum of a divergent series. This is a problem, because a divergent series can't have a finite sum. We must have started with a bad assumption, so
must not converge after all. It doesn't matter what an and bn are.
and 
converge, we can break up the series:
