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What is convergent and divergent? In sequences and series watch

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    Write down the first six terms of the following sequences, and decide which of the sequences appear to be convergnet and which appear to be divergent. For those which appear convergent, determine the limiting value to which they are tending.

    a) u_n = 2n+1
    = 3, 5, 7, 9, 11, 13

    e) u_n = \frac{1}{n}
    = 1, 1/2, 1/3, 1/4, 1/5, 1/6

    i) u_n = n(n+1)(n+2)
    = 6, 24, 60, 120, 210, 336
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    2n +1 is divergent
    1/n is converging to 0
    n(nplus1)(nplus2) js divergent
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    (Original post by anon1212)
    2n +1 is divergent
    1/n is converging to 0
    n(nplus1)(nplus2) js divergent
    Why though?
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    (Original post by Kash:))
    Why though?
    If a sequence is divergent the differences between terms either stays the same or gets bigger. in the case of 3,5,7,9 the difference is clearly the same. for the other divergent sequence the differences between terms is clearly getting bigger for each one.
    For a conergent sequence the differences between terms gets smaller moving through the sequence which is clearly evident from 1 to 1/2 to 1/3 etc...the sequence will keep on going till its 1/10000000000000000 to 1/infinity - the tiniest fraction, and is therefore converging towards zero.
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    OP: what level are you studying at?
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    Convergent is when the numbers come together.


    Divergent is when the numbers climax seperately.
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    (Original post by Kolya)
    OP: what level are you studying at?
    AS level, C2.
    So just the basics I guess
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    (Original post by anon1212)
    If a sequence is divergent the differences between terms either stays the same or gets bigger. in the case of 3,5,7,9 the difference is clearly the same. for the other divergent sequence the differences between terms is clearly getting bigger for each one.
    For a conergent sequence the differences between terms gets smaller moving through the sequence which is clearly evident from 1 to 1/2 to 1/3 etc...the sequence will keep on going till its 1/10000000000000000 to 1/infinity - the tiniest fraction, and is therefore converging towards zero.
    Not true, consider sqrt(1),sqrt(2),sqrt(3)....

    I can't think of a good way of explaining this to someone doing C2, but I'll give it a shot: Basically, if the sequence looks like it's getting bigger than any number you could possibly think of, we say it 'diverges to infinity'. Like 2n+1, well, if I guess 100 then you might say 'but putting n=100 in there gives me a bigger number!'
    Similarly, if it's the other way round and getting smaller (more negative) than any number you can imagine then it's probably diverging to -infinity. Like -1,-3,-5,.... will go to - infinity .
    If the number looks like it's getting closer and closer to another number- so close that however 'close' you want it to get it will eventually get that close...then we say it's 'converging' to that number . Like for example, 1/n^2, well the first few terms are 1/1, 1/4, 1/9, 1/16... hmm, looks like these denominators are getting bigger and bigger, and when that happens the number is getting closer and closer to 0. Infact, however close we want it, like, if we wanted it within 1/16 of 0, it'll eventually get that close! So it converges to 0.
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    Are you good at visualising things? Graphs?

    The graph of  y = \frac{1}{x} can be seen to be getting closer and closer to a finite point (in this case the x-axis -> so it's converging on 0)

    The graph of  y = x^2 is getting further and further away from any finite point.

    I hope this makes it a bit clearer...
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