Let f(x)=∑a_n(x^n) for |x|<R. If f(x)=f(-x) for all |x|<R, show that a_n=0 for all odd n.

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cosinusix | College Teacher | (Level 3) Assistant Educator

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If `f(x)=f(-x)` for all `|x|ltR, f(x)-f(-x)` is a constant 0 function for all `|x|ltR.`

 

The constant 0 function has a unique power series that is ` `

` `

Therefore all the coefficients of the power series of f(x)-f(-x)  ` ` are 0.

 

Now let's find the coefficients of f(x)-f(-x) as an expression of a_n

`f(x)-f(-x)=sum_0^(oo)a_nx^n-sum_0^(oo)a_n(-x)^n`

`f(x)-f(-x)=sum_0^(oo)(a_n-(-1)^na_n)x^n`

 

Hence` (a_n-(-1)^na_n)=0` for all n.

 

If n even `(-1)^n=1 ` and the condition becomes `a_n-a_n=0 ` which is true for any even n.

 

if n odd,` (-1)^n=-1` and the condition becomes `a_n+a_n=0. ` Therefore `a_n=0.`

 

In conclusion if `f(x)=f(-x)` , `a_n=0` for all odd n.

 

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