Limit comparison test on two same series? I am not quite sure how to phrase this question.
I have a series $$\sum_{n=1}^{\infty} |a_n|$$
I also have another series, for a natural number $N \geq 1$ $$\sum_{k=0}^{\infty}A_{k} = \sum_{n=N}^{\infty} |a_n|$$
Now, I am asked to write $\sum_{n=1}^{\infty} |a_n|$ in terms of $\sum_{k=0}^{\infty}A_{k}$
Since $\sum_{k=0}^{\infty}A_{k} = \sum_{n=N}^{\infty} |a_n|$, then (assuming I am correct) $$\sum_{n=1}^{\infty} |a_n| = \sum_{n=1}^{N} |a_n| + \sum_{k=0}^{\infty}A_{k}$$
Now the question asks me to show that one series converges if and only if the other series converges.
First thing I thought was the limit comparison test
If I do limit comparison test on $\sum_{n=1}^{\infty} |a_n|$ and $\sum_{k=0}^{\infty}A_{k}$, does the starting point matter?($n=1 \quad n=N$)
If not, this will end up as $$\lim_{n \rightarrow \infty} \frac{|a_{n}|}{|a_{n}|} = 1$$
And since the limit exists, both series will have the same result for convergence test.
My concern is that I am not sure the steps I took were legit.
We have in fact that
$$\sum_{n=1}^{\infty} |a_n| = \sum_{r=1}^{N-1} |a_r| + \sum_{k=0}^{\infty}A_{k}$$
or using limits of the corresponding seuqneces of partial limits:
$$\lim_{n\to\infty}\sum_{j=1}^n|a_j|=\sum_{r=1}^{N-1}|a_r|+\lim_{K\to\infty}\sum_{k=0}^KA_k$$
Observe that the first term in the right side is a finite number as it is a finite sum.
And since all the series (finite and infinite) are of positive terms, comparison and arithmetic of limits come in handy: if $\;\sum\limits_{k=0}^\infty A_k=T\;$ converges, then the sequence $\;\left\{\,\sum\limits_{j=1}^n|a_j|\,\right\}_{n=}^\infty\;$ is monotonic ascending and bounded above, by the number $\;\sum\limits_{r=1}^{N-1} |a_r|+A\;$ and it thus converges. and something similar the other way around: if $\;\sum\limits_{j=}^\infty |a_j|=S\;$ converges, then by arithmetic of limits
$$\lim_{K\to\infty}\sum_{k=0}^KA_k=\lim_{n\to\infty}\sum_{j=2}^n|a_j|-\sum_{r=1}^{N-1}|a_r|=S-\sum_{r=1}^{N-1}|a_r|$$
We have in fact that
$$\sum_{n=1}^{\infty} |a_n| = \sum_{r=1}^{N-1} |a_r| + \sum_{k=0}^{\infty}A_{k}$$
or using limits of the corresponding seuqneces of partial limits:
$$\lim_{n\to\infty}\sum_{j=1}^n|a_j|=\sum_{r=1}^{N-1}|a_r|+\lim_{K\to\infty}\sum_{k=0}^KA_k$$
Observe that the first term in the right side is a finite number as it is a finite sum.
And since all the series (finite and infinite) are of positive terms, comparison and arithmetic of limits come in handy: if $\;\sum\limits_{k=0}^\infty A_k=T\;$ converges, then the sequence $\;\left\{\,\sum\limits_{j=1}^n|a_j|\,\right\}_{n=}^\infty\;$ is monotonic ascending and bounded above, by the number $\;\sum\limits_{r=1}^{N-1} |a_r|+A\;$ and it thus converges. and something similar the other way around: if $\;\sum\limits_{j=}^\infty |a_j|=S\;$ converges, then by arithmetic of limits
$$\lim_{K\to\infty}\sum_{k=0}^KA_k=\lim_{n\to\infty}\sum_{j=2}^n|a_j|-\sum_{r=1}^{N-1}|a_r|=S-\sum_{r=1}^{N-1}|a_r|$$