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Convergent Series and Partial Sums 
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#1
Oct811, 08:03 AM

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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. 


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Oct811, 01:06 PM

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