Re: error on page 77 chapter 4.3 ?

What is written in the book is not S_n. It is the sum as n->infinity.

The series given is a geometric series:

a+ar+ar^2+ar^3+ar^4+....

where a=1 and r=x^2

S_n for a geometric series is:

S_n=a(1-r^n)/(1-r) equation (1)

which in this case where a=1 and r=x^2 is:

S_n=(1-x^(2n))/(1-x^2) equation (2)

However the limit of S_n as n->infinity is:

1/(1-r) provided that r<1

In this case 1/(1-x^2) provided that x^2<1

Quote:

Sn= 1+x^2+x^4+ .....+x^2n

sn*(1-x^2)= 1 + x^2 + x^4 + ..... + x^2(n-1) + x^2n

-x^2 - x^4 - x^6 - ..... - x^2(n) - x^(2(n+1))

sn*(1-x^2)= 1 - x^(2(n+1))

What you did above is correct except that in the first line the last term is x^2(n-1)

and then the last line becomes :

Sn*(1-x^2)=1-x^2n

which is the same as my equation 2.

The main thing that you have misunderstood is that in the book Penrose is talking about the infinite series and about the sum when n->infinity.

I hope this is helpful to you.

Vasco