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Everything else seems okay to me.
Yes, of course. Thank you.
Third line - in the denominator you have k1*k2*...*kn and you say below it "n times". It should be 1*2*...*kn and below it should be "kn times".
anonimnystefy, thank you. I see the mistake now
Therefore, series converges for k>=2
Well, the rest of his current work is okay. But that error is messing up the whole thing.
The second line is not correc(k(n+1))!=1*2*3*...*(k(n+1)-1)*(k(n+1))
Your first line is good.
Shouldn't manipulations maintain equality with the original assertion?
That does not?
I do not think there are mistakes:
Or are you talking about different equations?
Something is wrong right there.
The problem is:
For series to be convergent the next inequality should be true (by the Ratio Test):
Since we know that both k and n are positive we can omit absolute bars.
And now I simplify:
But since k is a constant this limit will never be less than 1. Therefore the series divergent for all possible k.
Did I make a mistake somewhere? Textbook is looking for a convergent series...