DinS          Written on 2018/4/27

1. What’s series?

Basically, when you add up all the terms in a sequence, you get a series. 

The problem is there’re infinite terms. 

Solution: partial sum. 

You’ll never finish adding all terms, but you can hope you’ll get closer to something as you add more terms.

So series is defined in terms of limit.

If , we say the series converges.

If , we say the series diverges.

Convergence is a major topic in series. We’ll cover it later.

Right now let’s see some typical series.

2. Geometric series

When you add up all terms in geometric progression you get geometric series.

See a sequence first. 

The sequence proceed like this: 

The question is 

To find the series, we must first solve partial sum. You can find a pattern here.

As to how this pattern is proved, you can look up a textbook on geometric progression. There’s a formula for it. Now all we care about is the series.

Thus we say:

We’ve solved a particular geometric series. Now we’ll solve generalized geometric series:

Notice we’re starting from 0. This can make proof easier and can be generalized later.

When r = 1, the series goes to infinite, countless 1 added up.

When r = -1, the series jumps between 1 and 0. The limit doesn’t exist.

In both cases, the series diverges.

Now let’s look at a generalized case.

S_n = r₀ + r₁ + r₂ + r₃ + … + r_n = r⁰ + r¹ + r² + r³ + … + rⁿ

If we multiply (1-r) on both sides we get:

(1-r)S_n = (1-r)( r⁰ + r¹ + r² + r³ + … + rⁿ)

After expanding right side we get:

(1-r)S_n = ( r⁰ + r¹ + r² + r³ + … + rⁿ) – ( r¹ + r² + r³ + r⁴ + … + rⁿ⁺¹)

Mass cancel is a cool thing. We get:

(1-r)S_n = r⁰ – rⁿ⁺¹

Since r is not 1 or -1, we can divide both sides by (1-r) and get:

We’ve worked out partial sum. Time for the series.

Since r is constant we can pull out all terms without n.
Now this should be clear.

If r > 1 or r < -1, the series goes to infinite. It diverges.

If -1 < r < 1, . We have S_n = 1/1-r.

In conclusion:

What if we start from k = m, not from 0?
We need a lemma, which I’ll not prove here.

Hopefully this makes sense.

Now that we already proved the case when k = 0, we can multiply r^m on both sides and get:

3. Telescoping series

A series is called telescoping series if we can apply the telescoping trick to it.

What is telescoping trick? See this.

This trick is called telescoping. If we split it into these, we can do mass cancel.

See an example here.

In general, if a function can be telescoped, we have

The trick of telescoping is very useful when we deal with sequences and series. If we can construct a telescoping pattern in our problem, we’re half done.

4. Harmonic series

This is a series of many stories, which we won’t go into detail here.

Let’s see a typical harmonic series and the trick of regrouping and underestimate.

 

Thus we can write the original formula to something like this.

We can group some terms together and know the group will be bigger than 1/2. Since there’ll be infinite counts of such group, the answer is the series diverges, even though 1/n gets very small.