Any complex number in the polar form can be expressed in Euler's form as cosθ+isinθ=eiθ. This form of the complex number is useful in finding the sum of series n∑r=0 nCr(cosθ+isinθ)r. n∑r=0 nCr(cosrθ+isinrθ)=n∑r=0 nCreirθ =n∑r=0 nCr(eiθ)r =(1+eiθ)n Also, we know that the sum of binomial series does not change if r is replaced by n−r. Using these facts, answer the following questions. The value of 100∑r=0 100Cr(sinrx) is equal to

Any complex number in the polar form can be expressed in Euler's form

1.	Any complex number in the polar form can be expressed in Euler's form as cosθ+isinθ=eiθ. This form of the complex number is useful in finding the sum of series n∑r=0 nCr(cosθ+isinθ)r. n∑r=0 nCr(cosrθ+isinrθ)=n∑r=0 nCreirθ =n∑r=0 nCr(eiθ)r =(1+eiθ)n Also, we know that the sum of binomial series does not change if r is replaced by n−r. Using these facts, answer the following questions. The value of 100∑r=0 100Cr(sinrx) is equal to
Answer» Any complex number in the polar form can be expressed in Euler's form as $cos θ + i sin θ = e^{i θ}$ . This form of the complex number is useful in finding the sum of series $n \sum r = 0^{n} C_{r} (cos θ + i sin θ)^{r}$ . $n \sum r = 0^{n} C_{r} (cos r θ + i sin r θ) = n \sum r = 0^{n} C_{r} e^{i r θ} = n \sum r = 0^{n} C_{r} (e^{i θ})^{r} = (1 + e^{i θ})^{n}$ Also, we know that the sum of binomial series does not change if $r$ is replaced by $n - r$ . Using these facts, answer the following questions. The value of $100 \sum r = 0^{100} C_{r} (sin r x)$ is equal to