Calculating the harmonic mean of $a,b$ problems with fractions

Question

Harmonic mean is $\frac{1}{\frac{\frac{1}{a}+\frac{1}{b}}{2}}=\frac{1}{\frac{\frac{b+a}{ab}}{2}}=\frac{2}{(b+a)(ab)}\neq \frac{2ab}{b+a}$

But the book says that the RHS is actually the harmonic mean where am I wrong? Because first the book specifies the harmonic mean with $\frac{1}{A(\frac{1}{a},\frac{1}{b})}$ where $A$ is the arithmetic mean but then the formulas don't match where is the error?

Answer 1

You made a mistake in step $3$.

$$\dfrac{1}{\frac{a+b}{\frac{ab}{2}}}=\dfrac{2ab}{a+b}=\text{HM}\{a, b\}=\dfrac{1}{\text{AM}\{1/a, 1/b\}}$$

Answer 2

It should be $$\frac{1}{\frac{\frac{1}{a}+\frac{1}{b}}{2}}=\frac{1}{\frac{a+b}{2ab}}=\frac{2ab}{a+b}.$$

Answer 3

Hint: It is $$\frac{2}{\frac{1}{a}+\frac{1}{b}}=\frac{2ab}{a+b}$$

Answer 4

Hint: The error is in your second equality. Note that $\frac{1}{\frac{\frac{A}{B}}{C}} = \frac{C}{\frac{A}{B}} = \frac{CB}{A}$.