Question about a tricky/difficult Taylor expansion of natural logarithm

AxiomOfChoice
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Can someone please tell me how to expand

<br /> \ln(x + \sqrt{1+x^2})<br />

for small x? I'd like to retain terms at least up to order x^5. Thanks!
 
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Let f(x) = ln( x + sqrt(1 + x^2) ).

Find f(0).
Find f '(0)
Find f ''(0)
Find f '''(0)
Find f ''''(0)
Find f '''''(0)

Then write f(x) = [f(0)x^0]/0! + [f '(0)x^1]/1! + [f ''(0)x^2]/2! + [f '''(0)x^3]/3! + [f ''''(0)x^4]/4! + [f '''''(0)x^5]/5!.

Does that sound right?
 

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