Constant Summation Rule
Simple overview of using summation notation to sum up a constant.
Constant Summation Formula
When summing a constant, I can use the following
\[\sum_{i=1}^{n} c = n\cdot c\]Here $c$ is an arbitrary constant.
It does not change as $i$ moves from $1$ to $n$.
In fact, using the summation notation the constant $c$ is added with itself $n$ times which is why the sum is equal to $nc$.
Watch this section: introduction at 0:00.
Expanding The Sum
The video then writes out what the summation means:
\[\sum_{i=1}^{n} c= \underbrace{c+c+c+\cdots+c}_{n\text{ times}}\]Since every term is still $c$, the long addition becomes multiplication:
\[\underbrace{c+c+c+\cdots+c}_{n\text{ times}} = n\cdot c\]We use the index of summation to help us understand how many times our constant is added together.
Watch this section: expanding the summation at 0:29.
Example: $n=20$ And $c=5$
Consider
\[\sum_{i=1}^{20} 5\]This means $5$ is added once for every value of $i$ from $1$ through $20$:
\[\underbrace{5+5+5+\cdots+5}_{20\text{ times added together}}\]There are $20$ copies of $5$, so the shortcut gives
\[\sum_{i=1}^{20} 5=20\cdot 5= 100\]Watch this section: example with $n=20$ and $c=5$ at 0:58.
Timestamp Guide
Related Calculus Work
For more calculus review, browse the Calculus notes.
If you are moving from sums into antiderivatives, the integration power rule page is a good next step.