How to decompose the below fraction into sum of 2 terms.

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The discussion focuses on decomposing the fraction \(\frac{1}{1+\frac{1}{x}}\) into a sum of two terms. The first step involves converting the fraction into an ordinary rational expression, resulting in \(\frac{x}{x(1 + \frac{1}{x})}\). Participants suggest splitting this expression into two components or recognizing that \(1\) can be rewritten as \(1 + \frac{1}{x} - \frac{1}{x}\) to facilitate the decomposition.

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as below, my math is getting rusty, hope some can help me out.

thanks.

[tex]\frac{1}{1+\frac{1}{x}}[/tex]
 
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First convert it into an ordinary rational expression (try simplifying

[tex] \frac x{x\left(1 + \dfrac 1 x\right)}[/tex]

then split the result into a sum of two pieces.

Alternately, notice that [itex]1 = 1 + \frac 1 x - \frac 1 x[/itex] and use that in the numerator.
 

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