- Given that
prove that
- Use Theorem 10.2 (that ), the result of part (a), and the fact that
to prove that
- Show that Theorem 10.2 when applied to
and
does not yield
Instead it gives us the formula
- Proof. (Not sure if I’m missing something deeper here…) Since we are given that
we have
- Proof. Given that
we apply Theorem 10.2 to this and the result of part (a) so,
- Proof. Applying Theorem 10.2 directly to the given expressions we have,
by 10.25, use -x to replace x to get 1-x+x^2-x^3…=1/(1+x). Then add 10.25 to this result to get (a)
For (a) and (b) you should use partial sums,s_n = the sum from k=0 to k=n-1 of a_k where a_k=(x^n+(-x)^n)/2 is the nth partial sum of 1+0+x^2+0+… and s_n = (1/2)t_n + (1/2)d_n where t_n and d_n are the nth partial sums of (10.25) and (10.28) and so s_n -> (1/2)(1/(1-x)+1/(1+x))=1/(1-x^2)
I guess the point of a) i is to show that their result is the same even though they are not identical (see exercise 24)
But I cannot think of any solution but the obvious ones