The usual definition of partial sum is a sum of the first m terms for some (finite) positive integer m. So if the terms are rational, then all partial sums are rational. As for the entire infinite sum, it can be rational or irrational.
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Odgovor korisnicima @wtgowers @johncarlosbaez i sljedećem broju korisnika:
To help you understand it, draw the curve in a Cartesian plane. My conditions specify it must be a reciprocal function y = 1/x where x is rational. Each term in the series is a point (x, y) in the curve. The points are apart and getting nearer to each other as x increases. This
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Odgovor korisnicima @Rey_Skywalker8 @johncarlosbaez i sljedećem broju korisnika:
Can you explain in more detail how each term in the series is a point (x,y) in the curve? On the face of it, a term in the series is a rational number and (x,y) is a pair of numbers, so they can't be equal. But maybe you aren't saying quite what you mean.
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Odgovor korisnicima @wtgowers @johncarlosbaez i sljedećem broju korisnika:
y is the term, y = 1/x x is the denominator. The point's coordinates are the term and its denominator.
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Odgovor korisnicima @Rey_Skywalker8 @johncarlosbaez i sljedećem broju korisnika:
In the case of the series 1 + 1/4 + 1/9 + 1/16 + ..., what are the points? Are they (1,1), (4,1/4), (9,1/9), (16,1/16), ... ? If so, they are not, contrary to your claim, getting closer and closer together -- quite the opposite.
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Odgovor korisnicima @wtgowers @johncarlosbaez i sljedećem broju korisnika:
This is my condition: dx/x = (Xn - Xn-1)/Xn → 0 x is denominator
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Odgovor korisnicima @Rey_Skywalker8 @johncarlosbaez i sljedećem broju korisnika:
In the case of the series 1 + 1/4 + 1/9 + ... what are the points on the curve that you say are getting closer and closer together?
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Odgovor korisnicima @wtgowers @johncarlosbaez i sljedećem broju korisnika:
dx/x → 0 difference of consecutive denominators with respect to x
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Odgovor korisnicima @Rey_Skywalker8 @johncarlosbaez i sljedećem broju korisnika:
In the case of the series 1 + 1/4 + 1/9 + ... what are the points on the curve that you say are getting closer and closer together? Asking again, because your "answer" didn't mention any points on the curve y=1/x. To make it easier, what are the first three points?
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Odgovor korisnicima @wtgowers @johncarlosbaez i sljedećem broju korisnika:
Your coordinates are correct. x-coordinate is denominator. dx/x = (Xn - Xn-1)Xn x-coordinates of points on the curve
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The points (1,1), (4,1/4), (9,1/9), ... are not getting nearer and nearer to each other as x increases. They are getting further and further apart. (It's true that their y-coordinates are getting nearer and nearer though -- is that what you meant?)
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Odgovor korisnicima @wtgowers @johncarlosbaez i sljedećem broju korisnika:
Use the formula dx/x = (Xn - Xn-1)/Xn It tends to 0 Yes, y = 1/x also tends to 0 Both tend to 0
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