Summation and Product Notation

Before working with infinite series, power series, or Taylor expansions, it pays to be fluent in the two compact notations that all of them are written in: summation (sigma, $\sum$) and product (pi, $\prod$). Both are just shorthand for “repeat this operation over a range of indices” — sums for $+$, products for $\times$.

Summation (Sigma) Notation

A sum is written as:

${\sum }_{n=a}^{b}f(n)=f(a)+f(a+1)+\cdots +f(b)$

• $n$ is the index of summation.
• $a$ is the lower limit (where the index starts) and $b$ the upper limit.
• $f(n)$ is the summand — the rule applied to each index.

When $b=\infty$ the sum is infinite, and whether it equals a finite value is exactly the question of convergence.

Reading a sum as a function

The key idea is that a sum whose summand depends on $x$ is itself a function of $x$. The geometric series is the canonical example:

$\frac{1}{1-x}={\sum }_{n=0}^{\infty }{x}^n=1+x+x^2+x^3+\dots (|x|<1)$

The left side is an ordinary function; the right side is the same function expressed as an infinite sum. Moving fluidly between the two forms is the whole game behind power series.

Useful manipulations

• Re-indexing (shifting): $\sum _{n=a}^{b}f(n)=\sum _{n=a+k}^{b+k}f(n-k)$ — shift the limits and compensate inside.
• Linearity: $\sum (\alpha f(n)+\beta g(n))=\alpha \sum f(n)+\beta \sum g(n)$.
• Splitting: pull out the first term(s) to change where a series starts, e.g. $\sum _{n=0}^{\infty }{a}_n=a_0+\sum _{n=1}^{\infty }{a}_n$.

Product (Pi) Notation

The multiplicative counterpart uses a capital pi:

${\prod }_{n=a}^{b}f(n)=f(a)\cdot f(a+1)\cdots f(b)$

Everything transfers over with $\times$ in place of $+$. The most important special case is the factorial, which is just a product of consecutive integers:

$n!={\prod }_{k=1}^{n}k=1\cdot 2\cdot 3\cdots n$

Factorials are why product notation matters for series at all — they appear in the denominators of Maclaurin and Taylor expansions.

Sum–product duality via logs

Logarithms turn products into sums, which is often how a product is evaluated:

$\log {\prod }_{n=a}^{b}f(n)={\sum }_{n=a}^b\log f(n)$

Why this is the foundation

Every series note in this garden is written in this notation:

• Series Convergence — when an infinite $\sum$ has a finite value.
• Power Series — a $\sum$ whose summand is a power of $x$.
• Taylor and Maclaurin Series — power series with factorial (a $\prod$) coefficients.

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Foundational note — the anchor the rest of the series cluster links back to.