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Math Help - Find the values of x for which the series converges, find the sum of those values.

  1. #1
    Junior Member Coop's Avatar
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    Question Find the values of x for which the series converges, find the sum of those values.

    Hi,

    Here's the question,

    "Find the values of x for which the series converges. Find the sum of the series for those values of x.
    summation(n=0,inf) [(x+3)^n]/2^n"

    I wasn't really sure how to approach it. I realized that it can become a convergent geometric series if |x+3| < 2, because that would make the absolute value of the r value less than 1.

    Taking this into account, I am proposing that -5< x < -1

    I am pretty sure am I right up to this point after looking around online, but I am confused about the second part. I thought maybe you could find the infinite sum when x is -5 and when x is -1 and then subtract the first value from the second value. Is this the correct way to go about it? I got 8/21 as an answer.

    Thanks
    Last edited by Coop; March 20th 2013 at 05:35 PM. Reason: grammar, typo
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  2. #2
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    Re: Find the values of x for which the series converges, find the sum of those values

    Quote Originally Posted by Coop View Post
    Hi,

    Here's the question,

    "Find the values of x for which the series converges. Find the sum of the series for those values of x.
    summation(n=0,inf) [(x+3)^n]/2^n"
    So \sum_{n=0}^\infty \left(\frac{x+ 3}{2}\right)^n

    I wasn't really sure how to approach it. I realized that it can become a convergent geometric series of |x+3| < 2, because that would make the absolute value of the r value = 1.
    Yes, that is a geometric sequence with "common ratio" \frac{x+ 3}{2} which must be between -1 and 1 in order that it converge absolutely.

    Taking this into account, I am proposing that -5< x < -1
    Okay, -1< \frac{x+ 3}{2}< 1 so that -2< x+ 3< 2, -5< x< -1.

    I am pretty sure am I right up to this point after looking around online, but I am confused about the second part. I thought maybe you could find the infinite sum when x is -5 and when x is -1 and then subtract the first value from the second value. Is this the correct way to go about it? I got 8/21 as an answer.

    Thanks
    I am afraid you have completely misunderstood this part of the question. It is asking for the sum as a function of x, not a difference between two values. You have correctly identified it as a geometric sequence, \sum_{n=0}^n ar^n, with common ratio r= \frac{x+3}{2} and first term a= 1. Now, recall that the sum of such a geometric series is \frac{a}{1- r}.
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  3. #3
    Junior Member Coop's Avatar
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    Re: Find the values of x for which the series converges, find the sum of those values

    Quote Originally Posted by HallsofIvy View Post
    I am afraid you have completely misunderstood this part of the question. It is asking for the sum as a function of x, not a difference between two values. You have correctly identified it as a geometric sequence, \sum_{n=0}^n ar^n, with common ratio r= \frac{x+3}{2} and first term a= 1. Now, recall that the sum of such a geometric series is \frac{a}{1- r}.
    Isn't the "a" value whatever you have when n=1, so wouldn't "a" be (x+3)/2 and since the whole sequence is in parenthesis, that would be the "r" value as well?
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    Re: Find the values of x for which the series converges, find the sum of those values

    No, the a value is the starting value. Your first term has n = 0, not n = 1.
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  5. #5
    Junior Member Coop's Avatar
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    Re: Find the values of x for which the series converges, find the sum of those values

    Quote Originally Posted by Prove It View Post
    No, the a value is the starting value. Your first term has n = 0, not n = 1.
    Oh of course, I forgot about the 0, thanks
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  6. #6
    Junior Member Coop's Avatar
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    Re: Find the values of x for which the series converges, find the sum of those values

    Quote Originally Posted by Prove It View Post
    No, the a value is the starting value. Your first term has n = 0, not n = 1.
    So the sum = 1/(1-((x+3)/2)), that's the final answer?
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  7. #7
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    Re: Find the values of x for which the series converges, find the sum of those values

    Well, do the algebra now! What does that reduce to?
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    Re: Find the values of x for which the series converges, find the sum of those values

    Quote Originally Posted by HallsofIvy View Post
    Well, do the algebra now! What does that reduce to?
    2/(x-1), right?

    Thanks for the help
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  9. #9
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    Re: Find the values of x for which the series converges, find the sum of those values

    Try again...
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  10. #10
    Junior Member Coop's Avatar
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    Re: Find the values of x for which the series converges, find the sum of those values

    Quote Originally Posted by Prove It View Post
    Try again...

    Yeah I forgot about the parenthesis :/ Anyway, thanks.
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