The great Swiss mathematician Leonhard Euler (biographyon p. 3) sometimes reached

Chapter 9, Problem 26

(choose chapter or problem)

The great Swiss mathematician Leonhard Euler (biography on p. 3) sometimes reached incorrect conclusions in his pioneering work on infinite series. For example,

Euler deduced that

\(\frac{1}{2}=1-1+1-1+\cdots\)

and

\(-1=1+2+4+8+\cdots\)

by substituting\(x = −1\) and \(x = 2\) in the formula

\(\frac{1}{1-x}=-x=1+x+x 2+x 3+\cdots\) What was the problem with his reasoning?

Equation Transcription:

Text Transcription:

1/2=1-1+1-1+ cdots

-1=1+2+4+8+ cdots

x=-1

x=2

1/1-x=-x=1+x+x2+x3+ cdots