Question
If $z_{k}=\cos \frac{\pi}{5^{k}}+i \sin \frac{\pi}{5^{k}} \quad k=1,2,3, \ldots$ then $z_{1} z_{2} z_{y}, \ldots \infty$ is(a) i(b) 0(c) $1+\mathrm{i}$(d) $\frac{(1+\mathrm{i})}{\sqrt{2}}$
Step 1
We need to find the product of all $z_{k}$ from $k=1$ to $\infty$. Show more…
Show all steps
Your feedback will help us improve your experience
Uma Kumari and 70 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
If $z=\cos (\pi / 3)-i \sin (\pi / 3)$ then $z^{2}-z+1=$ (a) $-2 \mathrm{i}$ (b) 2 (c) 0 (d) $-2$
If $z_{r}=\cos \frac{2 r \pi}{4}+i \sin \frac{2 r \pi}{4}$ where $r=0,1,2,3$ then $\frac{z_{0}+z_{1}}{z_{2}+z_{3}}$ equals (a) 1 (b) $-1$ (c) i (d) $-\mathrm{i}$
If $z=\sqrt{3}\left(\cos \frac{2 \pi}{3}+i \sin 2 \frac{\pi}{3}\right),$ express each of the following complex numbers in Cartesian form. a) $\frac{3}{\sqrt{3}+z}$ b) $\frac{2 z}{3+z^{2}}$ c) $\frac{3-z^{2}}{3+z^{2}}$
Complex Numbers
The complex plane
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD