Thinking of the common ratio as a variable x not as a

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: monic series in this context. Thinking of the common ratio as a variable x, not as a number r , we get ∞ xn = 1 + x + x2 + x3 + x4 + . . . + xn + . . . = f (x) = n=0 1 1−x (take a = 1 for simplicity). The series converges absolutely on (−1, 1) and on this interval its sum is 1 , so the series provides a series representation of a simple function from high school days. 1−x But this is calculus, so why not integrate both the Geometric series and its sum: x 0 x ∞ ∞ tn dt = n=0 n=0 0 ∞ tn dt = n=0 xn+1 = n+1 ∞ n=0 xn n 1 1 , dt = − ln(1 − x) = ln 1−x 0 1−t assuming that the integral of the infinite sum is the sum of the infinite number of integrals ; that’s x coming up too! Thus 1 x2 x3 x4 xn = x+ + + + ... + + ... , 1−x 2 3 4 n whi...
View Full Document

This note was uploaded on 06/05/2011 for the course MATH 408 D taught by Professor Gilbert during the Spring '11 term at University of Texas at Austin.

Ask a homework question - tutors are online