- #1
H12504106
- 6
- 0
Homework Statement
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_{2n-1} = a_n[/itex] and [itex]c_{2n} = b_n[/itex]. Prove that [itex]\sum_{n=1} c_n[/itex] converges.
Homework Equations
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_{2n-1}) = c_1 + c_2 +...+ c_{2n-1} = (a_1 +...+ a_n)+ (b_1 +...+b_{n-1}) = S_n +T_{n-1}[/itex]. Similarily, [itex](R_{2n}) = c_1 + c_2 +...+ c_{2n-1} + 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_{2n-1})[/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.