Pi (Product) Notation Calculator

∏ — Evaluate

Numeric product with term table, running product, log check

write — Expand

All factors listed; prime factorization of result

F(n) — Simplify

Auto-detects factorial, double factorial, shifted factorial patterns

∏∏ — Double

Nested product; inner first, outer second; separability

cancel — Telescope

Lists numerator/denominator values; finds and shows all cancellations

formula — Closed Form

Proves formula for ∏ as function of n, verifies numerically

ln∏ — Log Product

log(∏) = Σ log — converts product to sum; Stirling's note

A vs B — Compare

Ratio A/B, % difference, log-scale, product inequality rules

Definition \(\prod_{k=m}^{n}f(k)=f(m)\cdots f(n)\)

Empty product \(\prod_{k=m}^{n}=1\;\text{if }m>n\)

Factorial \(\prod_{k=1}^{n}k=n!\)

Partial fact. \(\prod_{k=m}^{n}k=n!/(m-1)!\)

Shifted fact. \(\prod_{k=m}^{n}(k+a)=(n+a)!/(m+a-1)!\)

Even nums \(\prod_{k=1}^{n}2k=2^n\cdot n!\)

Odd nums \(\prod_{k=1}^{n}(2k-1)=(2n-1)!!\)

Double fact. \((2n-1)!!=(2n)!/(2^n\cdot n!)\)

Constant \(\prod_{k=m}^{n}c=c^{n-m+1}\)

Power rule \(\prod_{k}f(k)^p=\left(\prod_k f(k)\right)^p\)

Product split \(\prod(f\cdot g)=\prod f\cdot\prod g\)

Telescope \(\prod_{k=m}^{n}\frac{f(k+1)}{f(k)}=\frac{f(n+1)}{f(m)}\)

Log identity \(\ln\prod f(k)=\sum\ln f(k)\)

AM-GM \(\left(\prod_{k=1}^n a_k\right)^{1/n}\leq\frac{1}{n}\sum a_k\)

Stirling's \(\ln(n!)\approx n\ln n-n+\tfrac{1}{2}\ln(2\pi n)\)

Double prod. \(\prod_i\prod_j f(i,j)=\prod_j\prod_i f(i,j)\)

∏Pi (Product) Notation Calculator