Calculus×Computing
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Power series

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

"Polynomials of infinite degree" ∑kck(x−a)k\sum_k c_k (x - a)^k. Inside their interval of convergence they can be added, multiplied, differentiated and integrated term by term — like polynomials.

Formulas

∑k=0∞ck(x−a)k,ddx∑ckxk=∑k ckxk−1\sum_{k=0}^{\infty} c_k (x - a)^k, \qquad \frac{\dd}{\dd x}\sum c_k x^k = \sum k\,c_k x^{k-1}
11−x=∑k=0∞xk(∣x∣<1)\frac{1}{1 - x} = \sum_{k=0}^{\infty} x^k \quad (|x| < 1)

Where it shows up in computing

  • Algorithm analysis and complexity★★★★★advancedScientific computing and algorithms

    Generating functions ∑anxn\sum a_n x^n solve recurrences and count structures in the analysis of algorithms.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

∿ Signals, media and vision

λ Scientific computing and algorithms

⚙ Robotics and control

ψ Quantum computing and physics

What depends on it

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

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