Paper 1, Section II, F

Analysis I | Part IA, 2012

(a) (i) State the ratio test for the convergence of a real series with positive terms.

(ii) Define the radius of convergence of a real power series n=0anxn\sum_{n=0}^{\infty} a_{n} x^{n}.

(iii) Prove that the real power series f(x)=nanxnf(x)=\sum_{n} a_{n} x^{n} and g(x)=n(n+1)an+1xng(x)=\sum_{n}(n+1) a_{n+1} x^{n} have equal radii of convergence.

(iv) State the relationship between f(x)f(x) and g(x)g(x) within their interval of convergence.

(b) (i) Prove that the real series

f(x)=n=0(1)nx2n(2n)!,g(x)=n=0(1)nx2n+1(2n+1)!f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}, \quad g(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}

have radius of convergence \infty.

(ii) Show that they are differentiable on the real line R\mathbb{R}, with f=gf^{\prime}=-g and g=fg^{\prime}=f, and deduce that f(x)2+g(x)2=1f(x)^{2}+g(x)^{2}=1.

[You may use, without proof, general theorems about differentiating within the interval of convergence, provided that you give a clear statement of any such theorem.]

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