Mathematics: Find a power series representation for f(x) = arctan(x).

Find a power series representation for f(x) = - id4143469, genyjoannerubiera

trevorpeterson20

12.01.2022

Recall that

$\dfrac{\mathrm d(\arctan x)}{\mathrm dx}=\dfrac1{1+x^2}$

and that for |x| , we have

$\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n$

So we get

$\displaystyle\frac1{1+x^2}=\frac1{1-(-x^2)}=\sum_{n=0}^\infty(-x^2)^n=\sum_{n\ge0}(-1)^nx^{2n}$

Taking the integral then gives

$\displaystyle\int\frac{\mathrm dx}{1+x^2}=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2n+1}+C$

We know that $\arctan0=0$ , so if we evaluate both sides of the equation above for x=0 , we would get a series that converges to 0, so that

$0=0+C\implies C=0$

and so

$\arctan x=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2n+1}$

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sjsmith21

02.11.2021

The number of dimes in Kiara's collection is 13 more than 5 times the

number of quarters in her collection. Kiara has 78 dimes in her

collection.

3) 78 = 5C+13

is the right answer.

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