Flashcards: Fundamental Power Series of Mathematical Functions — 12 cards

Exponential series — definition?

Series for e^x converging for all real x.

Hyperbolic sine series — expansion?

Sum of x^{2n+1}/(2n+1)! for all real x.

Hyperbolic cosine series — expansion?

Sum of x^{2n}/(2n)! for all real x.

Trigonometric sine series — expansion?

Sum of (-1)^n x^{2n+1}/(2n+1)! for all real x.

Trigonometric cosine series — expansion?

Sum of (-1)^n x^{2n}/(2n)! for all real x.

Power series intervals — key?

Convergence depends on radius and boundary points.

Logarithmic series for ln(1-x) — formula?

-∑_{n=1}^∞ x^n/n, for x in ]-1, 1[.

Logarithmic series for ln(1+x) — formula?

∑_{n=0}^∞ (-1)^n x^{n+1}/(n+1), for x in ]-1, 1[.

Geometric series — sum formula?

1/(1-x) = ∑_{n=0}^∞ x^n, for |x|<1.

Alternating geometric series — formula?

1/(1+x) = ∑_{n=0}^∞ (-1)^n x^n, for |x|<1.

Radius of convergence — meaning?

Distance from center where series converges.

Series convergence at boundaries — note?

Must check separately; may diverge or converge.