Suppose we are given a function \(\displaystyle{g{{\left({x}\right)}}}=\sum_{{{n}={1}}}{\frac{{{\sin{{\left({n}{x}\right)}}}}}{{{10}^{{n}}{\sin{{\left({x}\right)}}}}}},\ {x}\ne{k}\pi,{k}\in{\mathbb{{{Z}}}}\)

r1fa8dy5

r1fa8dy5

Answered question

2022-03-31

Suppose we are given a function
g(x)=n=1sin(nx)10nsin(x), xkπ,kZ

Answer & Explanation

kachnaemra

kachnaemra

Beginner2022-04-01Added 16 answers

It is a basic precalculus fact that for arbitrary complex z with |z|<1 one has
k=0zk=11z
Furthermore, by Euler's equation, for arbitrary real x we may write x=Imeix . Putting these two together we obtain
n=1sin(nx)10n=Imn=1(eix10)n
=Imeix101eix10
=Imeix(10eix)(10eix)(10eix)
=10sinx10120cosx
It follows that
g(x)=1010120cosx

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