Lesson 13 of 18Article20 min
Sequences, Series & Taylor Approximations
A sequence is a list aₙ. A series is the running sum ∑ aₙ. Convergence means partial sums approach a finite limit — not every infinite sum "makes sense."
Infinite sums with care
A sequence is a list aₙ. A series is the running sum ∑ aₙ. Convergence means partial sums approach a finite limit — not every infinite sum "makes sense."
- Geometric |r|<1 converges; |r|≥1 diverges.
- p-series ∑ 1/nᵖ converges for p>1.
- Alternating series can converge conditionally — absolute convergence is stronger.
Taylor polynomials
Near a point a, smooth functions look like polynomials built from f and its derivatives at a. More terms → better local approximation (within the radius of convergence for the infinite series).