Sums and Summation Notation

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Sigma Notation

Sigma notation, (also called summation notation) can be used to express sums of numbers.
For instance, the sum can be expressed as:
You can also use a function to express a sum of terms.
For instance, the sum of 5 consecutive perfect squares can be expressed as:
You can also use other functions.
Shown below are different functions you can use:
The sum of the first 7 reciprocals of cubes:
The sum of terms for

The sum of the first 4 terms of the arithmetic sequence
$Misplaced &\sum_{i=1}^4[5+2i] &= [5+2(1)] +[5+2(2)] +[5+2(3)] +[5+2(4)]\\ &=7 +9 +11 +13\\ &=40 \end{align*}$$$
In general, in expressing the sum of the first terms of any function into sigma notation, we do this:
We can also use the sigma notation to find the sum of all terms between the term to the term, not just the terms from the term to the term.
```
 $$\sum_{i=m}^n f(i) = f(m) +f(m+1)+f(m+2)+\cdots +f(n-1) +f(n)$$
```
For instance, we can use it to find sum of the terms from the term to the term.

Proof

We begin by setting a value for .
When
, we add exactly times.

If , we add c exactly times

In general, for any , we add exactly copies of .
Therefore, by simplifying the sum, we get:

$Missing \end{align*}\begin{align*}$

Thus, proving the property:

Proof

First, we start by expanding the summation notation.

Then using distributive property of addition, we factor a constant from the sum.

$Missing \end{align*}\begin{align*}$

Then, we express the resulting sum in sigma notation.

$Missing \end{align*}\begin{align*}$

Therefore, proving the property:

Proof

We begin by expanding the sum.

$Missing \end{align*}\begin{align*}$

Through associative property of addition, we can then group each term and each term.

$Missing \end{align*}\begin{align*}$

We then express each sum in sigma notation.

$Missing \end{align*}\begin{align*}$

Thereby, proving the property:

When proving the sums of differences, we expand and regroup like before.

$Missing \end{align*}\begin{align*}$

We can factor a on the terms.

$Missing \end{align*}\begin{align*}$

Then, we express each sum in sigma notation.

$Missing \end{align*}\begin{align*}$

Thereby, proving the property:

Combining these properties, we get: