## What is summation notation?

Summation notation is a way of writing a series such as

$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots+\frac{1}{n^2}+\cdots$

in a shorter form. The shorter form captures the fact that it is a sum by using the greek letter capital sigma $$\Sigma$$, indicates the index, in this case the letter $$n$$, the starting and ending values of the index $$n=1$$ and infinity $$\infty$$ (to indicate that the index increases one unit at a time without bound) and finally, the formula for the general term $$a_n=\frac{1}{n^2}$$.

$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots+\frac{1}{n^2}+\cdots=\sum_{n=1}^\infty\frac{1}{n^2}$

Summation notation is analogous to sequence notation $$\{a_n\}$$ with the exception that in sequence notation the sequence usually starts with $$n=1$$ and procedes indefinitely, so it is unnecessary to specify that $$n$$ goes from $$1$$ to infinity.

## Exercise 8.2.1

Rewrite the finite series ($$n$$ does not go to infinity) using summation notation: $$\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+\cdots+\dfrac{50}{51}$$

See Solution

## Exercise 8.2.2

Calculate the value of the sum:$$\quad\displaystyle\sum_{p=1}^4(p-1)^2$$

See Solution

## Exercise 8.2.3

Calculate the value of the sum:$$\quad\displaystyle\sum_{k=3}^7(k^2-2)$$

See Solution

## Exercise 8.2.4

Write the following as a single sum:$$\quad\displaystyle\sum_{p=0}^{3}p^2+\sum_{p=4}^{6}p^2$$

See Solution

## Exercise 8.2.5

Write the following as a single sum:$$\quad\displaystyle\sum_{p=0}^{3}p^2+\sum_{q=4}^{6}q^2$$

See Solution

## Defining functions using summation notation

Suppose $$f(x)=1-x+x^2-x^3$$. We could use summation notation and write

$f(x=\sum_{n=0}^3(-1)^nx^n$

Take special notice of how the factor $$(-1)^n$$ causes the sum to alternate in sign.

## Exercise 8.2.6

Write the function in summation notation: $$f(x)=1-\frac{1}{2}x+\frac{1}{3}x^2-\cdots+\frac{1}{101}x^{100}$$

See Solution