# Sigma Notation

## Key Questions

• First expand the series for each value of n $\frac{n}{2 n + 1}$=1/(2(1)+1+2/(2(2)+1+3/(2(3)+1+4/(2(4)+1+5/(2(5)+1

Next, perform the operations in the denominator...

$\frac{1}{3}$+$\frac{2}{5}$+$\frac{3}{7}$+$\frac{4}{9}$+$\frac{5}{11}$

Now, to add fractions we need a common denominator... in this case it's $3465$

Next, we have to multiply each numerator and denominator by the missing components...

$\frac{1}{3}$ gets multiplied by $1155$ giving $\frac{1155}{3465}$

(Divide the $3465$ by $3$ to get $1155$ and divide the rest by the given denominator.)

2/5*693/693=1386/3465, 3/7*495/495=1485/3465, 4/9*385/385=1540/3465 and 5/11*315/315=1575/3465

Now simply add the numerators together... $\frac{1155 + 1386 + 1485 + 1540 + 1575}{3465}$

giving $\frac{7141}{3465}$.

• $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = \frac{1}{2} ^ 1 + \frac{1}{2} ^ 2 + \frac{1}{2} ^ 3 + \cdots = {\sum}_{n = 1}^{\infty} \frac{1}{2} ^ n$

• Sigma notation can be a bit daunting, but it's actually rather straightforward. The common way to write sigma notation is as follows:

${\sum}_{x}^{n} f \left(x\right)$

Breaking it down into its parts:

• The $\sum$ sign just means "the sum".
• The $x$ at the bottom is our starting value for x. It usually has a number next to it: ${\sum}_{x = 0}$, for example, means we start at x=0 and carry on upwards until we hit...
• The $n$ at the top.
• The $f \left(x\right)$ is what we need to plug all these values into. At the end, we add the results obtained from here together, and that's our answer.

Note that it's not always $f \left(x\right)$ - it is most often $f \left(n\right)$ or $f \left(i\right)$.

As an example:

${\sum}_{x = 0}^{9} {\left(\sqrt{x} + 1\right)}^{2}$

means we need to find

${\left(\sqrt{0} + 1\right)}^{2} + {\left(\sqrt{1} + 1\right)}^{2} + {\left(\sqrt{2} + 1\right)}^{2} + \ldots + {\left(\sqrt{9} + 1\right)}^{2}$.