burkinaval1b

2022-01-04

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

The series:

$\sum _{n=1}^{\mathrm{\infty}}n{\left(\frac{1}{2i}\right)}^{n}$

The series:

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$\underset{n\to \mathrm{\infty}}{lim}\frac{{b}_{n+1}}{{b}_{n}}=\frac{1}{2}<1$

Conclusion: the given series converges absolutely hence converges

For this, we need to see if

as

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$\mathrm{\infty}\cdot 0$ 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.

Your reasoning about

Try the ratio test.

If you know about the geometric series

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

karton

Expert2022-01-11Added 613 answers

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