What is the power series for sqrt(x+1) using the square root algorithm?

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SUMMARY

The discussion focuses on deriving the power series for sqrt(x+1) using the square root algorithm. Participants mention the use of binomial expansion and the differentiation of the function, specifically noting that the derivative of sqrt(x+1) is 1/(2(x+1)^(1/2)). The conversation emphasizes the need to clarify which square root algorithm is being referenced, as multiple methods exist. The final expression discussed includes the integral representation and the application of inequalities for series expansion.

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  • Understanding of power series and their derivation
  • Familiarity with the square root algorithm and its variations
  • Knowledge of binomial expansion techniques
  • Basic calculus concepts, including differentiation and integration
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  • Research the specific square root algorithms applicable to power series
  • Study the binomial expansion method for functions like sqrt(x+1)
  • Learn about Taylor series and their applications in calculus
  • Explore inequalities used in series expansions and their implications
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lilcoley23@ho
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How would you go about finding the power series for sqrt(x+1) by applying the square root algorithm. I can do it using binomial expansion and other formulas but I'm not familiar with the square root algorithm involving variables.
 
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Which square root algorithm do you mean? There are several.
 
lilcoley23@ho said:
How would you go about finding the power series for sqrt(x+1) by applying the square root algorithm. I can do it using binomial expansion and other formulas but I'm not familiar with the square root algorithm involving variables.

[tex]\int\sqrt{x+1}\rightarrow \frac{2}{3}(x+1)^{\frac{1}{2}}[/tex]

It's just the usual 1/n+1x^n+1.

and nx^n-1

[itex]\frac{d}{dx} \sqrt {x+1} \rightarrow \frac{1}{2(x+1)^\frac{1}{2}}[/itex]

Which you can expand to a series using the inequality:

[itex](2x + r) r\leq a - x^2[/itex]
 
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