10,11}

Calculate the sums:

a) $1+a \cos \varphi+\ldots+a^{k} \cos k \varphi+\ldots(|a|<1)$

b) $a \sin \varphi+\ldots+a^{k} \sin k \varphi+\ldots(|a|<1)$

c) $\cos \varphi+C_{n}^{1} \cos 2 \varphi+\ldots+C_{n}^{n} \cos (n+1) \varphi$;

d) $\sin \varphi+C_{n}^{1} \sin 2 \varphi+\ldots+C_{n}^{n} \sin (n+1) \varphi$.
a), b) Let $z=a(\cos \varphi+i \sin \varphi)$. The required sums are the real and imaginary parts of the sum

$$
1+z+z^{2}+\ldots=\frac{1}{1-z}=\frac{1-\bar{z}}{|1-z|^{2}}=\frac{1-a \cos \varphi+i a \sin \varphi}{1-2 a \cos \varphi+a^{2}}
$$

c), d) Let $z=\cos \varphi+i \sin \varphi$. Then $1+z=2 \cos ^{2} \frac{\varphi}{2}+2 i \sin \frac{\varphi}{2} \cos \frac{\varphi}{2}=2 \cos \frac{\varphi}{2}\left(\cos \frac{\varphi}{2}+i \sin \frac{\varphi}{2}\right)$. The required sums are the real and imaginary parts of the expression

$$
\begin{gathered}
z(1+z)^{n}=2^{n} \cos ^{n} \frac{\varphi}{2}(\cos \varphi+i \sin \varphi)\left(\cos \frac{n \varphi}{2}+i \sin \frac{n \varphi}{2}\right)= \\
2^{n} \cos ^{n} \frac{\varphi}{2}\left(\cos \frac{(n+2) \varphi}{2}+i \sin \frac{(n+2) \varphi}{2}\right)
\end{gathered}
$$

## Answer

a) $\frac{1-a \cos \varphi}{1-2 a \cos \varphi+a^{2}}$
b) $\frac{a \sin \varphi}{1-2 a \cos \varphi+a^{2}}$
c) $2^{n} \cos ^{n} \frac{\varphi}{2} \cos \frac{(n+2) \varphi}{2}$
d) $2^{n} \cos ^{n} \frac{\varphi}{2} \sin \frac{(n+2) \varphi}{2}$.

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