Penjumlahan dan notasi pangkat

Summations

A summation, or a sum for short, is the sum of a list of numbers. For example, the sum of $1,2,3,4$ is $1+2+3+4=10$. We can write these sums in a more condensed way, by using something called sigma notation. The above example can be written as $$\sum_{i=1}^4 i=10$$

Let's break it down. First look at the bottom and top of the sigma. On the bottom we have something called the index, $i$. It says that $i=1$, which is where our index starts, and on the top there is a $4$, the ending point for the index. This means that we sum what's after the sigma (in the above example this is simply $i$) for all $i=1\to 4$, or $1+2+3+4$. Note that the index is incremented by one each time.

In general, $$\sum_{i=m}^nf(i)=f(m)+f(m+1)+\cdots +f(n-1)+f(n)$$

for $m\le n$

Sigma Operations

Let's look at the first example problem:

$$\displaystyle\sum_{i=1}^{10}(i+1)$$

We could have summed up all the values from $2$ to $11$, but we could have also done this:

$$\begin{align*}\sum_{i=1}^{10}(i+1)&=(1+1)+(2+1)+\cdots +(10+1)\\ &= (1+2+\cdots +10) +(1+1+\cdots +1)\\ &= \left(\sum_{i=1}^{10}i\right)+\left(\sum_{i=1}^{10}1\right)\end{align*}$$

We can separate a summation of a sum to the sum of the summation of each element of the sum. In other words,

$$\sum_{i=m}^n (f(i)+g(i))=\sum_{i=m}^n f(i)+\sum_{i=m}^n g(i)$$

Similarly,

$$\sum_{i=m}^n (f(i)-g(i))=\sum_{i=m}^n f(i)-\sum_{i=m}^n g(i)$$

From the first property of summations, if we replace $g(i)=f(i)$, then we have

$$\begin{align*}\sum_{i=m}^n 2f(i) &= \sum_{i=m}^n f(i)+f(i)\\ &=\sum_{i=m}^n f(i)+\sum_{i=m}^n f(i) \\ &= 2\sum_{i=m}^n f(i)\end{align*}$$

Thus, we can also factor out constants:

$$\sum_{i=m}^n cf(i)=c\sum_{i=m}^n f(i)$$

WARNING: these properties do not imply that

$$\sum_{i=m}^n f(i)g(i)=\left(\sum_{i=m}^n f(i)\right)\left(\sum_{i=m}^n g(i)\right)$$

nor does it mean that

$$\sum_{i=m}^n \dfrac{f(i)}{g(i)}=\dfrac{\sum\limits_{i=m}^n f(i)}{\sum\limits_{i=m}^n g(i)}$$

Products

A product has a similar idea to the summation, but instead of adding, we multiply. To express them in their compact form, we use pi notation. For example,

$$\prod_{i=1}^{10}i=1\cdot 2\cdot 3\cdots 9\cdot 10=10!$$

Similarly, to the sum, the product also has some properties:

$$\prod_{i=m}^nf(i)g(i)=\left(\prod_{i=m}^nf(i)\right)\left(\prod_{i=m}^ng(i)\right)$$ $$\prod_{i=m}^n\dfrac{f(i)}{g(i)}=\dfrac{\prod\limits_{i=m}^nf(i)}{\prod\limits_{i=m}^ng(i)}$$

See if you can prove why.

However, do note that $$\prod_{i=m}^n \left(f(i)\right)^c = \left(\prod_{i=m}^n f(i)\right)^c$$ and see if you can prove why.

A sum is the result of adding a sequence of values. For example, $1+4+9+16$ is the sum of the first four squares. We can write this sum compactly using sigma notation: $$\displaystyle\sum_{i=1}^4i^2.$$ The $\sum$ used here indicates that we add the values of $i^2$ for $i=1,2,3$ and $4$. In general, we can use sigma notation to write the sum of a sequence $a_m,a_{m+1},a_{m+2},\cdots,a_n$ as $$\displaystyle\sum_{i=m}^na_i = a_m + a_{m+1} + \cdots + a_n.$$

A product is the result of multiplying a sequence of values. Just like sigma notation for sums, there is a special way of writing the product of a sequence. Using pi notation, we can write $1\cdot2\cdot3\cdot4$ as $$\displaystyle\prod_{i=1}^4i.$$ More generally, the product of a sequence $a_m,a_{m+1},a_{m+2},\cdots,a_n$ is $$\displaystyle\prod_{i=m}^na_i$$

Summation notation

Sigma notation is a shorthand form used to denote a sum of many values. A summation is denoted by the capital Greek letter $\sum$ (sigma):

$$\begin{equation} \sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + ... + a_{n-1} + a_n \end{equation}$$

where $i$ is the index of summation, $i=1$ is the lower limit of the summation (starting point), $n$ is the upper limit (stopping point), and $a_i$ is the element being summed.

Product notation

Similarly, there is a shorthand form used to denote a product of many values. A product is denoted by the capital Greek letter $\prod$ (pi):

$$\begin{equation} \prod_{i=1}^{n} a_i = a_1 \times a_2 \times a_3 \times ... \times a_{n-1} \times a_n \end{equation}$$

where $i$ is the index of summation, $i=1$ is the lower limit of the product (starting point), $n$ is the upper limit (stopping point), and $a_i$ is the element being multiplied.

Sums

A summation is simply a shorthand for summing terms given by a formula together. These "sums" are things like $2+4+6+8$ or $1 + \frac 12 + \frac 14 + \cdots$ and can be expressed with a shorthand known as sigma notation: $$\displaystyle{\sum_{i=1}^4 2i = 2 + 4 + 6 + 8}$$ $$\displaystyle{\sum_{i=0}^\infty \dfrac 1{2^i} = 1 + \dfrac 12 + \dfrac 14 + \cdots}$$ This may appear confusing at first, but it's actually not that bad. Essentially, given a function $f(x)$, we can express the sum: $$f(j) + f(j+1) + \cdots + f(k)$$ As: $$\displaystyle{\sum_{i=j}^kf(i)}$$ (Note that $j$ and $k$ must be integers such that $k \ge j$, otherwise the sum is just $0$)

You might still be confused about the "$i=$" at the bottom. The purpose of $i$ is just to be a "dummy variable" used in the expression; we could've just replaced $i$ with $x$ or $z$ or anything. However, $i, j, k$ are the standard dummy variables that are used.

You might still be confused about the "$i=$" at the bottom. The purpose of $i$ is just to be a "dummy variable" used in the expression; we could've just replaced $i$ with $x$ or $z$ or anything. However, $i, j, k$ are the standard dummy variables that are used.

Summation Identities

These identities can be proven with basic arithmetic; they involve "taking out" or "splitting up" terms of the summation. $$\displaystyle{\sum_{i=j}^kcf(i) = c\sum_{i=j}^kf(i)}$$ $$\displaystyle{\sum_{i=j}^kf(i) + g(i) = \sum_{i=j}^kf(i) + \sum_{i=j}^kg(i)}$$ And for $j \le a < k$: $$\displaystyle{\sum_{i=j}^kf(i) = \sum_{i=j}^af(i) + \sum_{i=a+1}^kf(i)}$$

Ready for products? They're not as bad as they look as well!

Products

Products are just like sums, except they multiply the terms instead of adding them together. Products are expressed with big pi notation. As an example, the product $1 \cdot 3 \cdot 5 \cdot 7$ can be expressed as: $$\displaystyle{\prod _ {i=1}^4(2i-1)}$$

Product Identities

Like the summation identities discussed earlier, product identities behave the same way. Here they are: $$\displaystyle{\prod_{i=j}^kf(i)^c = (\prod_{i=j}^kf(i)})^c$$ $$\displaystyle{\prod_{i=j}^k(f(i) \cdot g(i)) = \prod_{i=j}^kf(i) \cdot \prod_{i=j}^kg(i)}$$ And for $j \le a < k$: $$\displaystyle{\prod_{i=j}^kf(i) = \prod_{i=j}^af(i) \cdot \prod_{i=a+1}^kf(i)}$$

That's it; I hope you enjoy your newfound knowledge of summations and products!

(Video) Sigma Notation

by Eddie Woo

Here's a video by Eddie Woo where he explains what sigma notation is and how you use it.

To recap, there's three ways to define a series or sequence.

1. Start listing them: $2,4,6,8,\cdot\cdot\cdot$
2. Algebraic expression for the $n^{\text{th}}$ term: $T_n=2n$
3. Recursive definition: $T_1=2;\ \ T_n=T_{n+1}+2$

Sigma Notation

Sigma notation relates to the second method above, algebraic expressions. Sigma is a Greek letter and we use the capital, $\Sigma$. It is a symbol we use to denote summation.

Suppose we wanted to find the sum of the even numbers. We could use sigma notation as a sort of short hand way to express this. It would be: $$\sum{2n}$$

Now, suppose we wanted to find the sum of the first ten even numbers. We have to define a starting point and an end point for our sigma. We want to start where $n$ is equal to 1. Since we want the first ten, we would end when $n$ equals 10. To denote this with sigma notation, it would look like: $$\sum_{n=1}^{10} 2n$$ If you write this out, it becomes: $$\sum_{n=1}^{10} 2n=2+4+6+8+10+12+14+16+18+20$$ Since this is an arithmetic series we can easily figure out the sum. $$\begin{align} \sum_{n=1}^{10} 2n&=\frac{n}2(a+l)\\ &=\frac{10}{2}(2+22)\\ &=5*22\\ &=110 \end{align}$$

Summation Notation

Summation notation helps us simplify how we write out sequences and series.

Sigma notation or summation notation helps us quickly and efficiently write out the pattern, or argument, for a sequence. Under the sigma, we see the lower limit, and on top of the sigma, we see an upper limit. When writing out the sequence, we start at the lower limit and apply the argument to that lower limit. We keep going until we hit the upper limit. We saw this when we learned about evaluating series.

An extension is product notation, designated by pi, $\pi$, instead of sigma, $\Sigma$.

Compared to summation notation, product notation is mainly used to express patterns in products. If there is some sort of pattern to a sequence of products, then we can use product notation.