Best solutions to - "Having trouble with this infinite series and deciding..."

burkinaval1b

Answered question

2022-01-04

Having trouble with this infinite series and deciding whether it converges or diverges.
The series:

\sum_{n = 1}^{\infty} n {(\frac{1}{2 i})}^{n}

Answer & Explanation

Bubich13

Beginner2022-01-05Added 36 answers

First let’s look if the series converges absolutely.
For this, we need to see if

\sum b_{n} = \sum \frac{n}{2^{n}}

converges. And this is immediate using the ratio test
as

lim_{n \to \infty} \frac{b_{n + 1}}{b_{n}} = \frac{1}{2} < 1

Conclusion: the given series converges absolutely hence converges

Chanell Sanborn

Beginner2022-01-06Added 41 answers

Hint. Your first thought is correct: look at the modulus.
Your reasoning about

\infty \cdot 0

is wrong.
Try the ratio test.
If you know about the geometric series

1 + x + x^{2} + \dots

you can differentiate, multiply by x and actually find out what your series converges to.

karton

Expert2022-01-11Added 613 answers

Consider $\sum_{n = 0}^{\infty} n z^{n}$ . The radius of convergence is $r = lim_{n \to \infty} \frac{1}{n^{\frac{1}{n}}} = 1$ . Since $| \frac{1}{2 i} | = \frac{1}{2}$ , the series converges.

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