Let [math]f[/math] be the function defined by

• Find the radius of convergence of the power series and also the domain of [math]f[/math].
• Write the derived series, and find its radius of convergence directly .
• Find the domain of the function defined by the derived series.

Let [math]f[/math] be the function defined by [math]f(x) = \sum_{i=1}^\infty \frac1{i2^i} (x-2)^i[/math], and follow the same instructions as in Problem Exercise.

Let [math]f[/math] be the function defined by [math]f(x) = \sum_{i=1}^\infty \frac1{\sqrt{i}} (x-2)^i[/math], and follow the same instructions as in Problem Exercise.

Let [math]f[/math] be the function defined by [math]f(x) = \sum_{i=0}^\infty \frac{(x-1)^i}{\sqrt{i+1}}[/math], and follow the same instructions as in Problem Exercise.

Find the domains of the functions [math]f[/math] and [math]g[/math] defined by the following power series.

• [math]f(x) = \sum_{i=1}^\infty \frac{(-1)^{i-1}} {(2i-1)!} x^{2i-1}[/math]
• [math]g(x) = \sum_{i=0}^\infty \frac{(-1)^{i}} {(2i)!} x^{2i}[/math].

If [math]f[/math] and [math]g[/math] are the two functions defined in Problem Exercise, show that

• [math]f^\prime (x) = g(x)[/math]
• [math]g^\prime (x) = - f(x)[/math].

Let [math]f[/math] and [math]g[/math] be the two functions defined in Problem Exercise (see also Problem Exercise).

• lab{9.7.7a} Evaluate [math]f(0)[/math], [math]f^\prime(0)[/math], [math]g(0)[/math], and [math]g^\prime(0)[/math].
• lab{9.7.7b} Show that [math]f[/math] and [math]g[/math] are both solutions of the differential equation [math]\frac{d^2y}{dx^2} + y = 0[/math].
• Write the general solution of the differential equation in \ref{ex9.7.7b}, and thence, using the results of part \ref{ex9.7.7a}, show that [math]f(x) = \sin x[/math] and that [math]g(x) = \cos x[/math].

Show as is claimed at the beginning of the proof of Theorem \ref{thm 9.7.2}, that it is a direct consequence of the Chain Rule that if this theorem is proved for [math]a=0[/math], then it is true for an arbitrary real number [math]a[/math].

Prove that every power series can be integrated, term by term. Specifically, prove the following two theorems.

• A power series [math]\sum_{i=0''^\infty a_i(x-a)^i[/math] and its integrated series
  [[math]] \sum_{i=0}^\infty \frac{a_i}{i+1} (x-a)^{i+1} [[/math]]
  have the same radius of convergence.}
• If the radius of convergence [math]\rho[/math] of the power series [math]\sum_{i=0''^\infty a_i(x-a)^i[/math] is not zero and if [math]f[/math] and [math]F[/math] are the functions defined, respectively, by
  [[math]] f(x) = \sum_{i=0}^\infty a_i(x-a)^i \quad \mbox{and} \quad F(x) = \sum_{i=0}^\infty \frac{a_i}{i+1} (x-a)^{i+1} , [[/math]]
  then
  [[math]] F(x) = \int f(x) \; dx + c . [[/math]]
  }

Starting from the geometric series

and the results of Problem Exercise, show that

for every [math]x[/math] such that [math]|x| \lt 1[/math].