(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 $\sum_{n=0}^{\infty} a_{n} x^{n}$.

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

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

(b) (i) Prove that the real series

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

have radius of convergence $\infty$.

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

[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.]