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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.
If we care about the value of a convergent series we need to specify the starting index. If we only care whether or not a series has a sum (that is, whether the series converges or diverges), we can ignore the limits of summation and write the series as
That's because the starting limit of summation doesn't affect whether a series converges or diverges.
If it sounds like we're waving our hands and pulling a lion out of a baseball cap, it might be worth looking at convergence of sequences. We know the tail of a sequence tells us about convergence, and the sequence of terms must go to zero for the series to converge. This means that we can ignore the head of the sequence, which happens to be the first terms of the series. The starting limit doesn't matter.
This means, an infinite series with general term an either converges or diverges, regardless of which term we start at, if
converges, so does
and vice versa.
This fact saves us effort and headaches. It saves us effort because instead of having to write
we can be lazy (or efficient) and write
It saves us headaches because we don't have to worry what we write for the starting limit of summation.
Suppose we're asked to show that
converges, but it's easier to show that
converges. We can wave our hands, do the one that's easier, and say, "since the starting limit of summation doesn't affect whether the series converges or diverges,
Be Careful: The choice of starting limit of summation doesn't affect whether a series converges or diverges. However, it will affect the final sum of the series. If you're asked to find a sum (not just to say whether the sum exists), you do have to make sure your starting limit is correct.