I was given the following function:
$$ f(x) = x + \frac{2x^3}{1\cdot3} + \frac{2\cdot4x^5}{1\cdot3\cdot5} + \frac{2\cdot4\cdot6x^7}{1\cdot3\cdot5\cdot7}… $$ $$ \forall x \in [0,1) $$
And then I was asked to find the value of $ f(\frac{1}{\sqrt{2}}) $, which obviously requires me to compute the closed form expression of the infinite series.
I tried 'Integration as a limit of sum' but I was unable to modify the expression accordingly. How do I approach the problem?
Best Answer
$${\frac {\arcsin \left( x \right) }{\sqrt {1-{x}^{2}}}}=x+{\frac {2}{ 3}}{x}^{3}+{\frac {8}{15}}{x}^{5}+{\frac {16}{35}}{x}^{7}+O \left( {x} ^{9} \right) $$ Then $$f\left(\frac{1}{\sqrt2}\right) =\frac{\arcsin\left(\frac{1}{\sqrt2}\right)}{\sqrt{1-\frac12}}=\frac{\pi\sqrt2}{4}$$