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Math Help - Calc II: Finding a sequence from a partial sum equation

  1. #1
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    Calc II: Finding a sequence from a partial sum equation

    I don't even know where to start for this question. If someone could help me with the first few steps, I would be very grateful!

    If the nth partial sum of a series Sigma (infinite sum, n=1) is s(sub)n=3-n^(-n), find a(sub)n and the partial sum of a(sub)n
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  2. #2
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    Re: Calc II: Finding a sequence from a partial sum equation

    Start at the beginning.

    What is a partial sum? What is meant by s_n\,?
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    Re: Calc II: Finding a sequence from a partial sum equation

    A partial sum is taking all integers (to infinity) plugging them into an equation, and adding each term together. I started with a table, and tried to find some sort of pattern by subtraction between each term, but got nowhere fast.
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    Re: Calc II: Finding a sequence from a partial sum equation

    Quote Originally Posted by clarinetqueen92 View Post
    A partial sum is taking all integers (to infinity) plugging them into an equation, and adding each term together. I started with a table, and tried to find some sort of pattern by subtraction between each term, but got nowhere fast.
    Gee, I thought the partial sum was the result of adding the first n terms of the series

    s_n=a_1+a_2+a_3+\dots+a_{n-2}+a_{n-1}+a_n=\sum_{k=1}^n\,a_k

    or was I wrong about that?

    If I'm right, then a_1=s_1=3-(1^{-1})=2
    a_2=s_2-s_1=3-(2^{-2})-2=3-\frac{1}{4}-2

    etc.
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    Re: Calc II: Finding a sequence from a partial sum equation

    Ok, our variables are a bit different. On your sigma, my n=infinity, k=1, and a(sub)k=3-k2^(-k). The question asks me to find a(sub)k and the sum from k=1 to infinity of a(sub)k.

    (Sorry, I don't know how to use online math notations at all.)

    Edit: Wait a minute, I missed some of your post. Hold on.

    Edit 2: Ok, so we agree on our definitions of partial sums, but this is an infinite sum, and I don't understand any more about the problem than what I started out with... Sorry if I'm being dense...
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    Lightbulb Re: Calc II: Finding a sequence from a partial sum equation

    do you want to find \sum_{k=1}^{\infty}(3-k2^{-k})?

    If yes then you should take \lim_{n\rightarrow\infty}\sum_{k=1}^{n}(3-k2^{-k}) =\lim_{n\rightarrow\infty}2^{-n} (3 \times 2^n n+n-2^{n+1}+2)

    Can you make something out of it?
    Last edited by sbhatnagar; October 17th 2011 at 09:56 PM.
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    Re: Calc II: Finding a sequence from a partial sum equation

    Quote Originally Posted by clarinetqueen92 View Post
    Ok, our variables are a bit different. On your sigma, my n=infinity, k=1, and a(sub)k=3-k2^(-k). ...
    Is that a typo in your a_k ? Does that 2 belong there?
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    Member sbhatnagar's Avatar
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    Re: Calc II: Finding a sequence from a partial sum equation

    I found that the series \sum_{k=1}^{\infty}(3-\frac{k}{2^k}) never converges but \sum_{k=1}^{\infty}\frac{k}{2^k}=2.
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