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Math Help - Convergence question (Calc)

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    Convergence question (Calc)

    For the series below, I have to prove that it either converges, diverges, or that whether it converges/diverges cannot be determined. I'm not sure how to go about this, so any help would be appreciated!

    Sigma starting at j=1 and ending at infinity of (2/3)^j + (3/4)^j

    I mean, I know each of the individual terms in the series converge because they are geometric. (2/3)^j converges to 3 and (3/4)^j converges to 4. Does this mean I can just add them to get 7 and because of this, it converges?
    Last edited by clockingly; April 10th 2007 at 10:08 AM.
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    Quote Originally Posted by clockingly View Post
    For the series below, I have to prove that it either converges, diverges, or that whether it converges/diverges cannot be determined. I'm not sure how to go about this, so any help would be appreciated!

    Sigma starting at j=1 and ending at infinity of (2/3)^j + (3/4)^j

    I mean, I know each of the individual terms in the series converge because they are geometric. (2/3)^j converges to 3 and (3/4)^j converges to 4. Does this mean I can just add them to get 7 and because of this, it converges?
    Hello,

    the proposed way to do the problem is OK, because both summands have the same exponent - but since

    ∑[from 1 to ∞](2/3)^j = 2 and

    ∑[from 1 to ∞](3/4)^j = 3

    the series converges against 5 (or do you say to 5 (?)).
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