**1. Sigma Notation.** A sequence is an ordered set. We will deal with finite sequences. We write

(1)

Now, we can form a *series* by

(2)

Some terminology:

(3)

We call the “” the sigma notation (it’s a capital sigma, , for “sum”) and we call the small subscript in an *index*. So what are some examples?

Example 1Consider . What is its sum? It is

(4)

Note the difference betweenpowers, e.g. , andsubscripts(e.g. ).

Example 2Another series

(5)

which has 4 terms.

Example 3A simpler example

(6)

This is a toy example we will expand upon later.

Now, note we can rewrite the same sum several different ways.

We can change the index

(7)

without any problem.

Finite sums are called **“Finite Series”**.

There are some rules for simplifying sums:

**Sum Rule:** suppose we have two sequences , . The sum can be rearranged as

(8)

**Difference Rule:**

**Constant Multiple Rule:** Let be a constant, then

(9)

**Constant Sum Rule:** Let , be constants, then

(10)

Note that the constant sum rule implies the others. For example, when , we get the sum rule; when and we get the constant multiple rule; and when and we get the difference rule.

**2. Gauss’ Sum.** Let be any positive integer. Then

(11)

How can we prove this?

Suppose were even. Then writing out the sum:

(12)

We can rearrange the terms writing

(13)

So we can group these into pairs:

(14)

How many such pairs exist? There are such pairs, each summing to . Thus for even , the sum is

(15)

What about odd ?

Write . Then we have

(16)

Using our previous result, we can rewrite the right hand side as

(17)

Observe by simple arithmetic we have

(18)

Thus we have proven Gauss’ formula.

**3. Another Proof of Gauss’ Sum.** We can use a technique called *Mathematical Induction* which is a three-step procedure.

**Base Case:** we prove Gauss sum works for some nice value of . We pick . The sum is trivial

**Inductive Hypothesis:** we suppose that Gauss’ formula works for any :

But we’re not done yet! We need to prove that when changes to Gauss’ formula still works. This is the next step…

**Inductive Case:** we write the formula out and try to connect it back to the inductive hypothesis:

We see that we have some constant plus a series which resembles the inductive hypothesis. We use the inductive hypothesis to rewrite our series as

Using basic arithmetic, we simplify this to be

This is precisely Gauss’ formula for .

This concludes the inductive proof for Gauss’ formula.

**Exercise 1.** Prove by induction

**Exercise 2.** Prove by induction

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