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For $f(x) = \log_2(2x)$

Derive the truncated Taylor series expansion and the respective error term when truncating the Taylor series for f(x + h) developed at x = 1 after the n-th term.

Any idea, hint or solution is welcomed.

Thank you

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  • $\begingroup$ Please show your work. Update your question with the Taylor expansion formula, and try to apply the formula to your specific function. After that, write what problems you are having and we will be happy to guide you. $\endgroup$ – gt6989b Feb 13 '17 at 0:10
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Hint:

$$\log_2(2x)=1+\log_2(e)\ln(x)$$

Now use well known expansions of $\ln(x)$ at $x=1$.

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  • $\begingroup$ Is this the well know expansion of ln(x)? $\sum^\infty_{n=0}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$ If yes, how do I continue from here? $\endgroup$ – zeeks Feb 13 '17 at 0:29
  • $\begingroup$ @zeeks Usually, I'd say this:$$\ln(x)=\int_0^{x-1}\frac1{1+x}\ dx$$And to recommend you see the geometric series. The second alternative is Google images. And the third, what I would call brute force, would be to take derivatives repeatedly. $\endgroup$ – Simply Beautiful Art Feb 13 '17 at 0:31

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