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Convergent Series and Partial Sums

by H12504106
Tags: convergent, partial, series, subsequence, sum
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H12504106
#1
Oct8-11, 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_{2n-1} = 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_{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.
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LCKurtz
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Oct8-11, 01:06 PM
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Quote Quote by H12504106 View Post
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_{2n-1} = a_n[/itex] and [itex]c_{2n} = b_n[/itex]. Prove that [itex]\sum_{n=1} c_n[/itex] converges.


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_{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.
The assertions you make look to be all true. But I think you need to give a more complete explanation for the last two sentences, because proving it carefully is essentially the same as the original problem. I would think along the lines if Σ an = S and Σbn = T, you should be able to show directly that the c series converges to S + T with an ε, N argument.


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