# 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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Teacher / professor, points: 2.98K

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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.

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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