Question

What is importance of infinite series?

Answer 1

Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis.

Convergent Geometric Series: For −1<x<1:

(1)

∑i=0∞xi=1+x+x2+...+xi+...=11−x

Derivative of Convergent Geometric Series: For −1<x<1:

(2)

∑i=0∞ixi−1=1+2x+3x2+...+ixi−1+...=1(1−x)2

Antiderivative of Convergent Geometric Series: For −1≤x<1:

(3)

∑i=0∞xi+1i+1=x+x22+x33+...+xi+1i+1+...=−ln(1−x)

Exponential Series: For all x∈R:

(4)

∑i=0∞xii!=1+x+x22!+x33!+...+xii!+...=ex

(5)

∑i=0∞(−1)ixii!=1−x+x22!−x33!+...+(−1)ixii!+...=e−x

(6)

∑i=0∞x2i(2i)!=1+x22!+x44!+...+x2i(2i)!+...=ex+e−x2=coshx