What is the sum from n=0 to infinity of (0.5+0.5i)^n ?

The sum from n=0 to infinity of (0.5+0.5i)^n is converges

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sum from n=0 to infinity of (0.5+0.5i)^n

n = 0 \sum \infty (0.5 + 0.5 i)^{n}

converges

Solution steps

n = 0 \sum \infty (0.5 + 0.5 i)^{n}

Multiply by the conjugate of

0.5 + 0.5 i : \frac{0.5}{0.5 - 0.5 i}

= n = 0 \sum \infty (\frac{0.5}{0.5 - 0.5 i})^{n}

Simplify

(\frac{0.5}{0.5 - 0.5 i})^{n} : (\frac{1 + i}{2})^{n}

= n = 0 \sum \infty (\frac{1 + i}{2})^{n}

Apply Series Ratio Test:

converges

= converges

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What is the sum from n=0 to infinity of (0.5+0.5i)^n ?
The sum from n=0 to infinity of (0.5+0.5i)^n is converges