How the book diverges

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It is about the determination of the \ (x \ in \ mathbb {R} \) for which the series (absolute) converge or diverge? I know how to examine a given series for convergence (trivial criterion minor / majorant criterion, root / quotient criterion). But it is not clear to me how I approach such a task. I don't want to understand the solution, I want to understand the mindset behind it.

I have now found, among other things, that you can also work classically with the root or quotation criterion and that it then has to be <1. I tried the first row: For \ (a_n = \ left (\ frac {1} {x- \ frac {1} {n}} \ right) ^ n \) I come up with the root criterion
$$ \ sqrt [n] {\ left | \ left (\ frac {1} {x- \ frac {1} {n}} \ right) ^ n \ right |} = \ frac {1} {\ left | x- \ frac {1} {n} \ right |} $$ and thus to
$$ \ lim_ {n \ to \ infty} \ sqrt [n] {\ left | a_n \ right |} = \ lim_ {n \ to \ infty} \ frac {1} {\ left | x- \ frac {1} {n} \ right |} = \ frac {1} {\ left | x \ right |} $$.
$$ \ frac {1} {\ left | x \ right |} <1 \ Longrightarrow \ left | x \ right | > 1 $$
So, according to my considerations, the series would converge on the intervals \ ((- \ infty, -1), (1, \ infty) \). Is that true, and if so, can someone help me clear up the margins? Just grit my teeth on it.
asked
\ (((n (n + 1)) ^ {\ frac {1} {3}}) ^ {\ frac {1} {n}} \ to 1 \)
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gerdware
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That's a very good approach. You say you can handle normal series but not this one. Imagine these rows here as normal rows that just happen to have a parameter (\ (x \)) where others would have a number.
In the first row you did everything right so far. From this you can conclude that it converges on \ ((- \ infty, -1) \ cup (1, \ infty) \) and diverges on \ ((- 1,1) \), so it only remains, \ (x \ in \ {- 1,1 \} \) to investigate. You can use that in each case and get a "normal" power series that you can examine for convergence. Let's start with \ (x = 1 \), it's a bit easier: Inserted the series is $$ \ sum_ {n = 1} ^ \ infty \ frac1 {\ left (1- \ frac1n \ right) ^ n} . $$ The limit value in the denominator should look familiar to you: \ (\ lim_ {n \ to \ infty} \ left (1- \ frac1n \ right) ^ n = \ frac1e \) (if not, then ask again ), so the summands do not converge to \ (0 \) and the series diverges. For \ (x = -1 \) it goes something similar, here you should first consider \ (\ lim_ {n \ to \ infty} \ left | \ left (-1- \ frac1n \ right) ^ n \ right | \).
In the second row, first check for which \ (x \) the individual summands converge to \ (0 \) at all.
In the third row, the root criterion is again suitable (as almost always when there is a ^ n somewhere). Here you have to examine the edges again, see if you can do that. Otherwise you are of course welcome to ask again.
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