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Math Help - Series Help (I don't see how this adds up is all)

  1. #1
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    Series Help (I don't see how this adds up is all)

    I'm trying to figure out how my teacher solved this infinite series, the problem is:

    The sum as k = 1 goes to infinity of (1/3)^k,
    so the Sequence of terms is: an = (1/3)^n,
    and the Sequence of partial sums is S1 = 1/3, S2 = (1/3 + 1/9), S3 = etc.

    Then my teacher goes to say that: Lim as n -> Infinity of Sn = 1/2 , and I have no idea how he got that, can anyone help me figure this out?
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  2. #2
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    Quote Originally Posted by vandorin View Post
    I'm trying to figure out how my teacher solved this infinite series, the problem is:

    The sum as k = 1 goes to infinity of (1/3)^k,
    so the Sequence of terms is: an = (1/3)^n,
    and the Sequence of partial sums is S1 = 1/3, S2 = (1/3 + 1/9), S3 = etc.

    Then my teacher goes to say that: Lim as n -> Infinity of Sn = 1/2 , and I have no idea how he got that, can anyone help me figure this out?

    The sum of a FINITE geometric series is a+aq+aq^2+\ldots +aq^{n-1}=\sum^{n-1}_{k=1}aq^k=a\frac{q^n-1}{q-1} .

    Now, if |q|<1 then  q^n\xrightarrow [n\to\infty]{}0 , so passing to the limit in the first sum we get:

    \sum^\infty_{k=0}aq^k=\lim_{n\to\infty}a\frac{q^n-1}{q-1}=-\frac{a}{q-1}=\frac{a}{1-q} .

    Now check again your exercise, ony taken into account that your infinite sum begins with k=1 and NOT with k=0 .

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    The sum of a FINITE geometric series is a+aq+aq^2+\ldots +aq^{n-1}=\sum^{n-1}_{k=1}aq^k=a\frac{q^n-1}{q-1} .

    Now, if |q|<1 then  q^n\xrightarrow [n\to\infty]{}0 , so passing to the limit in the first sum we get:

    \sum^\infty_{k=0}aq^k=\lim_{n\to\infty}a\frac{q^n-1}{q-1}=-\frac{a}{q-1}=\frac{a}{1-q} .

    Now check again your exercise, ony taken into account that your infinite sum begins with k=1 and NOT with k=0 .

    Tonio
    Now it makes sense! Thank you!
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