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Math Help - need help with limit proof

  1. #1
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    need help with limit proof

    I'm having trouble thinking this out.
    I have this problem:
    Suppose that for all x in (−c, c) the identity
    a0 + a1x + .... + an-1x^(n-1) + (an + A(x))x^n = b0 + b1x + .... + bn-1x^(n-1) + (bn + B(x))x^n, where the limit as x goes to 0 of A(x) = lim as x goes to 0 of B(x) = 0. How would I show that a0 = b0, a1 = b1, ...... , an = bn. It seems to make sense, but I'm not quite show sure how to show it. How would you do it? Thanks for any help on this.
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  2. #2
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    Quote Originally Posted by MKLyon View Post
    I'm having trouble thinking this out.
    I have this problem:
    Suppose that for all x in (−c, c) the identity
    a0 + a1x + .... + an-1x^(n-1) + (an + A(x))x^n = b0 + b1x + .... + bn-1x^(n-1) + (bn + B(x))x^n, where the limit as x goes to 0 of A(x) = lim as x goes to 0 of B(x) = 0. How would I show that a0 = b0, a1 = b1, ...... , an = bn. It seems to make sense, but I'm not quite show sure how to show it. How would you do it? Thanks for any help on this.
    a_0 + a_1x+...+ (a_n +A(x))x^n = b_0+b_1x+...+(b_n+B(x))x^n
    Take the limit x\to 0 of both sides, and we get,
    a_0=b_0.
    Subtract these from both sides to get,
    a_1x+...+(a_n+A(x))x^n = b_1x+...+(b_n+B(x))x^n
    Divide by x\not = 0 to get,
    a_1+...+(a_n+A(x))x^{n-1}=b_1+...+(b_n+B(x))x^{n-1}.
    Take the limit again,
    a_1 = b_1
    Keep on repeating this argument.
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  3. #3
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    That's really clever. Thank you. One other question:
    If the polynomial p(x) = the sum from k = 0 to n of akx^k for all x in (-c,c), what would be the coefficients ak?

    Thanks again for the help.
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  4. #4
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    Quote Originally Posted by MKLyon View Post
    That's really clever. Thank you. One other question:
    If the polynomial p(x) = the sum from k = 0 to n of akx^k for all x in (-c,c), what would be the coefficients ak?

    Thanks again for the help.
    For any polynomial p(x) we can write p(x) = \sum_{n=0}^{\infty} \frac{p^{(n)}(0)}{n!}x^n*.



    *)This is actually a finite sum because eventually the derivative is zero. Take for example p(x) = 1+x+x^2 then p'''(x) = 0 so p^{(n)}(x) = 0 for all n\geq 3.
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  5. #5
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    If the polynomial (I'll write out the sum in long form) is:
    a0 + a1x^1 + .... + anx^n and it equals zero, doesn't it mean all the coeffcients ak must be zero? Is this the answer to my second question?
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  6. #6
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    Quote Originally Posted by MKLyon View Post
    If the polynomial (I'll write out the sum in long form) is:
    a0 + a1x^1 + .... + anx^n and it equals zero, doesn't it mean all the coeffcients ak must be zero? Is this the answer to my second question?
    A polynomial f(x) which is non-zero has at most \deg f(x) zeros. So if f(x) is always zero on an interval then it must be the zero polynomial because it can only have finite number of zeros.
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