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Math Help - Analytic function

  1. #1
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    Analytic function

    Prove that 2^x is analytic on R and find its Maclaurin expansion.

    thanks in advance
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  2. #2
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    Verify for yourself that: f^{(n)}(x) = \left(\ln 2\right)^n 2^x

    So the Maclaurin series of f(x) = 2^x is given by:  \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!} x^n = \sum_{n=0}^{\infty}\frac{\left(\ln 2\right)^n}{n!} x^n

    This series converges for all x (use the ratio test to verify).

    To prove that f(x) = 2^x is equal to the sum of its Maclaurin series, we must show that \lim_{n \to \infty} R_{n}(x) = 0 where R_n(x) is the remainder, i.e.:

    R_{n}(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1} = \frac{(\ln 2)^n2^c}{(n+1)!}x^{n+1} where c \in (0, x)

    This fact should come in handy: \lim_{n \to \infty} \frac{x^n}{n!} = 0

    With this, we've shown f(x) is equal to its Taylor series over the reals and thus analytic.
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