Sigma Notation

The central idea

The notation $\sum_{i=m}^{n}a_i$ means $a_m+a_{m+1}+\cdots+a_n$. The index $i$ is a dummy variable, $m$ and $n$ are bounds, and there are $n-m+1$ terms. Finite sums are linear: constants factor out and sums or differences may be separated.

Definitions, hypotheses, and notation

Index shifts are best verified by generated terms. For example, $\sum_{i=1}^{n}a_i=\sum_{j=0}^{n-1}a_{j+1}$: both begin with $a_1$ and end with $a_n$. Changing only the lower bound or only the subscript creates a different list. The letter used for a dummy index has no meaning outside its sum.

Growth estimates provide checks on closed forms. A sum of $n$ terms each roughly size $n^p$ should have order $n^{p+1}$. Thus $\sum i$ grows quadratically and $\sum i^2$ cubically. Exact identities can be proved by induction or geometric arguments and then used to evaluate Riemann-sum limits without expanding every term.

Conceptual meaning

Sigma notation compresses repeated addition while retaining the pattern and bounds. It is the language of Riemann sums, sequences, numerical approximation, and probability. Changing a dummy index without changing the generated terms does not change the sum.

A dependable method and decision rule

1. Read the lower and upper bounds and count terms inclusively.
2. Substitute successive index values to expand a short sum.
3. For formulas, use linearity and standard identities such as $\sum_{i=1}^ni=n(n+1)/2$.
4. When shifting an index, change the expression and both bounds consistently.
5. Check the first and last terms after every reindexing.

Fully worked example

Graphical or geometric meaning

The partial sums $1,4,9,16$ form square numbers. Geometrically, each odd number adds an L-shaped border to the previous square, explaining $\sum_{i=1}^{n}(2i-1)=n^2$.

Common mistakes and why they fail

Verification and reasonableness checks

• Expand the first two and last term.
• For small $n$, compare a formula with direct addition.
• Estimate the sign and growth order before simplifying.

Check the index before using a formula

A finite sum is determined by its summand and both bounds. Changing the starting index adds or removes terms. Expand the first two and last terms before applying identities such as $\sum_{i=1}^n i=n(n+1)/2$. Constants factor out and sums distribute over addition, but nonlinear operations do not: the square of a sum is generally not the sum of squares. Reindexing is valid only when the summand and bounds all shift consistently. If $j=i-1$, replace every occurrence of $i$ and transform both endpoints. Testing a small value of $n$ is a quick way to expose an indexing error before simplifying the general case.

Practice

1. Expand $\sum_{k=2}^{5}k^2$.
2. How many terms are in $\sum_{j=3}^{10}a_j$?
3. Find $\sum_{i=1}^{10}i$.

Connections and next steps

• Riemann Sums
• Sequences
• Infinite Series