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I am trying to find the Taylor Series representation of $f(x)= \sqrt x + \frac1{\sqrt x}$ at $a = 1.$ With $5$ terms.

I know how to get the series expansion. centered at $a=1$. with $5$ terms… However I can't figure out how to get the series representation.

I know that the $\sqrt x$ Taylor Series is — $(x-1)^n/n!$

José Carlos Santos
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supersyd
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  • You need to explain your question better. You know how to get the 5 terms, but you don’t know how to get the series of 5 terms? It almost sounds like you’re saying you can’t figure out how to add :). Surely you mean something else. – Erick Wong May 08 '21 at 17:47
  • I'm trying to get the Taylor Series representation. I know how to get the expansion using 5 terms but the question is asking for the representation! – supersyd May 08 '21 at 17:50
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    “ I know that the $\sqrt x$ Taylor Series is --- $(x-1)^n/n!$” This is wrong. Not clear what you mean to say here, but this is inaccurate. – Thomas Andrews May 08 '21 at 17:51
  • It’s also unclear what you mean by “series presentation.” If you know how to get the series expansion, how is that different from the series representation? – Thomas Andrews May 08 '21 at 17:52
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    @supersyd As far as I can tell, the two terms are interchangeable. What is missing that you need to get a Taylor representation? Is your question really just “what’s the difference between the two concepts?”. If yes, please state so clearly because it has nothing to do with this particular function. – Erick Wong May 08 '21 at 17:53
  • Simply add both series with the same index for $x^{1/2}$ and $x^{-1/2}$. Use the binomial series for both. – Тyma Gaidash May 08 '21 at 17:53

1 Answers1

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The first five terms are:

$$\begin{align}f(1)&+\frac{f’(1)}{1!}(x-1)^1\\&+\frac{f’’(1)}{2!}(x-1)^2\\&+\frac{f’’’(1)}{3!}(x-1)^3\\&+\frac{f’’’’(1)}{4!}(x-1)^4\end{align}$$

So you just need to compute the $f(1),f’(1),\dots, f’’’’(1).$

Thomas Andrews
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