We didn't use our psychic powers to figure out how the sequence of partial sums of a series is related to the terms of the series. Take a look at our first "mathic" trick.
"If the sequence of partial sums converges, then the sequence of terms must converge to zero." Why shouldn't things work out this way? Start with a series
Suppose the sequence of partial sums converges to some number L. In symbols,
The partial sums are
S1 = a1
S2 = a1 + a2
Sn – 1 = a1 + a2 + ... + an – 1
Sn = a1 + a2 + ... + an – 1 + an
To get from Sn – 1 to the next partial sum Sn, we add on one more term, an. If the partial sums Sn – 1 and Sn are close together, we must not be adding on very much to get from Sn – 1 to Sn.
In order for the partial sums to be getting closer and closer, we must be adding on smaller and smaller amounts as the index approaches ∞. This means the values an must be getting closer and closer to zero.
In limit notation,
Tada! Hopefully, this is a convincing argument that, if the sequence of partial sums Sn converges, then the sequence of terms an must converge to zero. If you like pictures better than formulas, check out the pictures that go with this argument.
What about our other "mathic" trick?
"If a series converges, its terms must converge to 0."
Start with a series, written
The partial sums are defined as usual by
Sn = a1 + ... + an.
If the series converges, then its partial sums converge to some finite value L:
Then it's also true that
since this expression gives the limit of the same sequence.
Since both these limits exist, we can subtract them.
Look at each side of this equation separately.
The left-hand side is
For the right-hand side, the nth partial sum is the sum of the first n terms of the series:
Sn = a1 + a2 + ... + an – 1 + an
The (n – 1)st partial sum is the sum of the first (n – 1) terms:
Sn = a1 + a2 + ... + an – 1
When we subtract the partial sums, all the terms except an cancel out:
Sn – Sn – 1 = (a1 + a2 + ... + an – 1 + an) – (a1 + a2 + ... + an – 1)
Replacing both sides of the equation
Alakazam! No, not the Pokemon. The terms an converge to 0.
We'd like to point out one extra thing. If a series converges, the sequence of partial sums converges to some finite number L:
We just proved that if this happens, then the terms must converge to 0:
Unless L is 0, these two sequences will converge to different values. That's fine. The sequence of partial sums and the sequence of terms don't need to converge to the same place.
Be Careful: The converse of this fact is not true. If the terms of a series converge to 0, that does not mean the series must converge.
It's uber important that we understand this next example. If it doesn't make sense on first reading, read it again. If it still doesn't make sense, ask a teacher or friend for help.
V.I.E.: Very Important Example
Look at the harmonic series again.
The terms of this series definitely converge to 0:
However, as we'll prove later, the harmonic series, as a series, diverges.
We can almost guarantee that any exam about series will have a question like this:
True or False: If the terms of a series converge to 0, then that series converges.
Or maybe like this:
Prove that the following statement is true or provide a counterexample: If the terms of a series converge to 0, then the series converges.
The statement is false, because of the Very Important Example. Our counterexample is the harmonic series, a series whose terms converge to 0 but which does not converge as a series.