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What would be the simplest way to do this?

Hanul Jeon
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Billy Thompson
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    Go ahead, write out what you want, compute derivarives, the works. You are supposed to learn something by doing this, you know... – vonbrand Apr 07 '13 at 02:16

3 Answers3

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First, let $z:=x-16$, then this is the same as $\sqrt{16+z}$ at $z=0$. Use the binomial series: $$(1+x)^\alpha=\sum_{n\ge 0}\binom\alpha n x^n$$ Now we have $$(16+z)^{1/2}=4\cdot\left(1+\frac z{16}\right)^{1/2}=\\ =4\cdot\sum_{n\ge 0} \binom{1/2}n\frac{z^n}{16^n}\,.$$

Berci
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The simplest way? Perhaps, Taylor expansion for f(x) = sqrt(x) centered at a=16.

Mark McClure
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    This may be slightly misleading: this "simplest way", besides not contributing a lot to some newbie's understanding, would hardly be acceptable in any more or less decent college/university, so it may not be the wisest thing to do (for the student, of course). – DonAntonio Apr 07 '13 at 02:17
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    @DonAntonio Honestly, the question is so ridiculous, that it's hard to give a serious answer. As a college professor, I'd certainly expect even an average student to be able to do this with modest effort. Also as a college professor, I tell students to use all kinds of tools to understand things and I think that WolframAlpha is a bit under-utilized, as illustrated by the fact that I typed his exact title into WolframAlpha. – Mark McClure Apr 07 '13 at 02:35
  • Is this answer any worse than the fully worked out solutions to elementary HW questions that we see on this site all the time? This is a serious question. I honestly dont' know how to deal with this issue. – Mark McClure Apr 07 '13 at 02:35
  • Well @Mark: even full-worked solution leave a little window of hope for the student to understand a little: (s)he has to copy the question and, hopefully, understand it. WA doesn't care, and cannot either, about this. I also think this is a lame question from a college and up student, but... – DonAntonio Apr 07 '13 at 02:38
  • @Mark: I don’t downvote, but in my opinion it is indeed much worse: it teaches no mathematics at all, and I can’t see it doing anything to increase anyone’s understanding of Taylor series. Nor do I think it legitimate to justify a shoddy answer by ridiculing the question. Your answer would have been reasonable as a mildly facetious comment. – Brian M. Scott Jun 02 '13 at 11:10
  • @BrianM.Scott I can assure you that my answer was neither facetious nor intended to ridicule. I believe that low effort questions should receive low effort answers, in part, because low effort questions fail to convey the context necessary to formulate a full response. To often, I've put effort into an answer only to have the OP say, "Well, I really meant...". I also genuinely believe that WA is an underutilized resource and one of the areas that I try to cover in my answers here is its proper use. Although, again, proper use requires effort. – Mark McClure Jun 02 '13 at 13:56
  • @BrianM.Scott One other comment - it is clearly not the purpose of this site to teach mathematics. If it were, then fully worked out solutions 10 minutes after the question is asked (like the accepted answer to this question) would be excluded. – Mark McClure Jun 02 '13 at 13:59
  • @Mark: In my view it should have been facetious, and I did not suggest that it was intended to ridicule the OP, just the question. And as I pointed out, you don’t know when it’s asked that this is a low-effort question. It is a perfectly reasonable question for someone who knows the textbook approach to ask: it’s not exactly unheard of in mathematics for there to be more than one approach, some easier than others. // If the point is understanding of Taylor series, this is obviously not a proper use of W|A. – Brian M. Scott Jun 02 '13 at 18:45
  • @Mark: On the contrary, Berci’s answer is a good teaching answer: it shows an efficient way to calculate that Taylor series that is not the textbook approach. – Brian M. Scott Jun 02 '13 at 18:46
  • @BrianM.Scott Hmm... The Binomial theorem certainly is in my textbook. :) If Berci had started with "Hint: Look into the binomial theorem" and if his answer had then come about via some dialogue, then I could certainly agree with you. However, he simply typed out the correct a fully correct answer a scant 10 minutes after the question was posted. You can find any number of complaints about this approach to "teaching" on Meta, as I'm sure you know. – Mark McClure Jun 02 '13 at 18:52
  • let us continue this discussion in chat – Mark McClure Jun 02 '13 at 18:55
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note that $f(x)=2^0x^{1/2}$, $f'(x)=-2^1 x^{-1/2}$, $f''(x)=\frac{2^2}{3} x^{-3/2}$,$f'''(x)=-\frac{2^3}{15}x^{-5/2}$ $\cdots$

Then just plug these into the normal formula for a taylor series expansion

formla for taylor series

here your $a=16$. Then see what patterns you can find to write these in summation notation!

MITjanitor
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  • (+1) because this is the way it should be done on homework (which is my guess here). However, it is not what I would call "simpler" – robjohn Apr 07 '13 at 14:53