Register to reply 
Convergent Series and Partial Sums 
Share this thread: 
#1
Oct811, 08:03 AM

P: 6

1. The problem statement, all variables and given/known data
Let [itex]\sum_{n=1} a_n[/itex] and [itex]\sum_{n=1} b_n[/itex] be convergent series. For each [itex]n \in \mathbb{N}[/itex], let [itex]c_{2n1} = a_n[/itex] and [itex]c_{2n} = b_n[/itex]. Prove that [itex]\sum_{n=1} c_n[/itex] converges. 2. Relevant equations 3. The attempt at a solution Not sure whether the following solution is correct or not. Let [itex]S_n, T_n, R_n[/itex] be the partial sums of the series [itex]\sum_{n=1} a_n, \sum_{n=1} b_n, \sum_{n=1} c_n[/itex] respectively. Now [itex](R_{2n1}) = c_1 + c_2 +...+ c_{2n1} = (a_1 +...+ a_n)+ (b_1 +...+b_{n1}) = S_n +T_{n1}[/itex]. Similarily, [itex](R_{2n}) = c_1 + c_2 +...+ c_{2n1} + c_{2n} = (a_1 +...+ a_n)+ (b_1 +...+b_n) = S_n +T_n[/itex]. Since [TEX]\sum_{n=1} a_n[/itex] and [itex]\sum_{n=1} b_n[/itex] converges, the sequence [itex](S_n)[/itex] and [itex](T_n)[/itex] converges. Since [itex](R_{2n1})[/itex] and [itex](R_{2n})[/itex] converges to the same value, [itex](R_n)[/itex] converges. Hence, the series [itex]\sum_{n=1} c_n[/itex] converges. 


#2
Oct811, 01:06 PM

HW Helper
Thanks
PF Gold
P: 7,658




Register to reply 
Related Discussions  
Plotting partial sums for Fourier Series in matlab  Engineering, Comp Sci, & Technology Homework  0  
Geometric series partial sums question  Calculus & Beyond Homework  1  
Show that the partial sums of a power series have no roots in a disk as n>infty  Calculus & Beyond Homework  3  
Sequences, Cumulative Sums, Partial Sums Plot  Ti89 Titanium  Calculators  0  
Series (Partial Sums )  Calculus & Beyond Homework  2 