Sigma notation is a compact way to write a finite or infinite sum. This calculator focuses on finite sums,
where the lower and upper limits are fixed integers.
1. Meaning of sigma notation
\[
\sum_{k=m}^{N} f(k)
\]
means:
\[
f(m)+f(m+1)+f(m+2)+\cdots+f(N)
\]
The letter \(k\) is called the index of summation. Other common index letters are \(i\), \(n\), \(j\), and \(m\).
2. Example expansion
Consider:
\[
\sum_{k=1}^{10}(k^2+3k)
\]
This means:
\[
(1^2+3\cdot1)+(2^2+3\cdot2)+\cdots+(10^2+3\cdot10)
\]
Using standard formulas:
\[
\sum_{k=1}^{10}(k^2+3k)
=
\sum_{k=1}^{10}k^2
+
3\sum_{k=1}^{10}k
\]
\[
=
\frac{10(11)(21)}{6}
+
3\cdot\frac{10(11)}{2}
\]
\[
=
385+165=550
\]
3. Important power-sum formulas
| Sum |
Formula |
| \(\sum_{k=1}^{N}1\) |
\(N\) |
| \(\sum_{k=1}^{N}k\) |
\(\dfrac{N(N+1)}{2}\) |
| \(\sum_{k=1}^{N}k^2\) |
\(\dfrac{N(N+1)(2N+1)}{6}\) |
| \(\sum_{k=1}^{N}k^3\) |
\(\left(\dfrac{N(N+1)}{2}\right)^2\) |
4. Linearity of summation
Summation is linear. This means sums can be split apart:
\[
\sum_{k=m}^{N}\left(f(k)+g(k)\right)
=
\sum_{k=m}^{N}f(k)
+
\sum_{k=m}^{N}g(k)
\]
Constants can also be pulled outside:
\[
\sum_{k=m}^{N}c\,f(k)
=
c\sum_{k=m}^{N}f(k)
\]
5. Geometric sums in sigma form
A geometric sum can be written with sigma notation:
\[
\sum_{n=1}^{N} ar^{n-1}
\]
The finite geometric sum is:
\[
\sum_{n=1}^{N} ar^{n-1}
=
a\frac{1-r^N}{1-r}
\qquad (r\ne1)
\]
6. Alternating sums
An alternating sum often uses a factor such as:
\[
(-1)^k
\quad\text{or}\quad
(-1)^{k+1}
\]
For example:
\[
\sum_{k=1}^{N}\frac{(-1)^{k+1}}{k}
=
1-\frac12+\frac13-\frac14+\cdots
\]
7. Factorials in sigma notation
Factorials also appear in many sums:
\[
\sum_{n=0}^{N}\frac{1}{n!}
\]
This partial sum approaches \(e\) as \(N\) grows:
\[
\sum_{n=0}^{\infty}\frac{1}{n!}=e
\]
8. Graph interpretation
The graph in the calculator shows two pieces of information:
- The bars show individual term values \(f(k)\).
- The line shows the running partial sum.
If term bars are positive, the running sum increases. If terms are negative, the running sum decreases.
Alternating signs make the running sum move up and down.
9. Common mistakes
- Do not confuse the index variable with the upper limit.
- Do not forget to include both endpoints of the sum.
- \(\sum_{k=1}^{10}\) has \(10\) terms, not \(9\).
- The index letter is a dummy variable; \(\sum_{k=1}^{10}k^2\) and \(\sum_{i=1}^{10}i^2\) mean the same thing.
- Check arithmetic carefully when expanding terms by hand.
