Solution 19.

Solution.   Since      converges for  ,  

the series converges absolutely for  ,  where .

      To fully justify this statement would take quite an effort, so it will suffice to say that the proof would be
      
similar to the proof in Theorem 7.4 (in Section 7.2) where we discussed uniform convergence of Taylor series.

Therefore, if   ,   then the series      converges.

          Since   ,    

for all  ,  and   

                       converges,

the Weierstrass M test implies that   converges uniformly on  .

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell