Mathematical Series You Should Know
Mathematical Series You Should Know
Mathematical series representations are very useful tools for describing images or solving imaging problems. They may be used to expand a function into terms that are individual monomial expressions (powers) of the coordinate.
Geometric Series
Adjacent terms in a geometric series exhibit a constant ratio. If the scale factor for adjacent terms is \( t \), the series has the form:
\[ \sum_{n=0}^{\infty} t^n \]
If \( |t| < 1 \), this converges to:
\[ \sum_{n=0}^{\infty} t^n = \frac{1}{1 – t} \]
Example:
\[ (0.9)^{-1} = \frac{1}{0.9} = \frac{1}{1 – 0.1} = 1 + 0.1 + 0.01 + 0.001 + \dots = 1.1111\dots \]
Finite Geometric Series
The finite geometric series including \( N + 1 \) terms:
\[ \sum_{n=0}^{N} t^n = \frac{1 – t^{N+1}}{1 – t} \quad \text{if} \quad |t| < 1 \]
Example:
\[ \sum_{n=0}^{4} (0.1)^n = 1 + 0.1 + 0.01 + 0.001 + 0.0001 = 1.1111 \]
Binomial Expansion
The binomial expansion is given by:
\[ (1+x)^n = \sum_{r=0}^{\infty} \binom{n}{r} x^r \]
Where the binomial coefficient is defined as:
\[ \binom{n}{r} = \frac{n!}{(n-r)!r!} \]
1 + x2
1 + x2
