AP EAMCET · Maths · Complex Number
The value of \(\sum_{n=0}^{\infty}\left(\frac{2 i}{3}\right)^n\) is
- A \(\frac{9+6 i}{13}\)
- B \(\frac{9-6 i}{13}\)
- C \(9+6 i\)
- D \(9-6 i\)
Answer & Solution
Correct Answer
(A) \(\frac{9+6 i}{13}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \sum_{n=0}^x\left(\frac{2 i}{3}\right)^n \\ &=1+\left(\frac{2 i}{3}\right)+\left(\frac{2 i}{3}\right)^2+\left(\frac{2 i}{3}\right)^3+\ldots . \\ &=\frac{1}{1-\frac{2 i}{3}}=\frac{3}{3-2 i} \times \frac{3+2 i}{3+2 i}\end{aligned}\)
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