Real Analysis:assistance?
I have a few questions to ask:
1) Prove that Card(ℤ>0) ≠ Card(ℝ).
2)Give an example of a sequence (a(superscript n)) such that none of inf (a(superscript n)), lim inf (a(superscript n)), lim sup (a(superscript n)), and sup(a(superscript n)) are equal.
3)Find the power series expansions and the radius of convergence of e^x , log(1 + x) ,1/(1−x), (1 + x)^(1/2), arctan x, and sinh x.
Once again thankyou for your help it is very much appreciated :)
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1) Since ℤ>0 is countable (and thus equivalent to ℤ), it suffices to show that ℝ is uncountable. This is usually done by place ℝ in bijection to (0,1). Showing (0,1) is uncountable uses a standard Cantor Diagonal Argument.
2) Try a sequence like {1 - 1/2, -1 + 1/3, 1 - 1/4, -1 + 1/5, ...}
3) Power series:
(i) e^x = sum(k=0 to infinity) x^k/k!, R = infinity
(ii) 1/(1-x) = sum(k = 0 to infinity) x^k, R = 1.
(iii) By (ii) replacing x with -t yields
1/(1 + t) = sum(k = 0 to infinity) (-1)^k t^k, R = 1.
Integrating from 0 to x yields
ln(1 + x) = sum(k = 0 to infinity) (-1)^k x^k/k, R = 1.
(iv) By (ii),replacing x with -t^2 yields
1/(1 + t^2) = sum(k = 0 to infinity) (-1)^k t^(2k), R = 1.
Integrating from 0 to x yields
arctan x = sum(k = 0 to infinity) (-1)^k x^(2k+1)/(2k+1), R = 1.
(v) sinh x = (1/2)(e^x + e^(-x))
= (1/2) [sum(k=0 to infinity) x^k/k! + sum(k=0 to infinity) (-1)^k x^k/k!]
= sum(k = 0 to infinity) x^(2k)/(2k)!, R = infinity
(vi) Using the Binomial Theorem with exponent 1/2,
(1 + x)^(1/2) = 1 + sum(k = 1 to infinity) C(1/2, k) x^k, R = 1.
where C(1/2, k) = (1/2)(-1/2)(-3/2)...(1/2 - k + 1) / k!.
Good luck!