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.
Teste tes connaissances avec un QCM de 6 questions sur Fundamental Power Series of Mathematical Functions.
1. What is the exponential series?
2. What is the power series expansion of the hyperbolic sine function, sinh(x), as given in the course content?
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