Having trouble with this infinite series and deciding whether it

burkinaval1b 2022-01-04 Answered
Having trouble with this infinite series and deciding whether it converges or diverges.
The series:
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{n}{\left({\frac{{{1}}}{{{2}{i}}}}\right)}^{{n}}\)

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Expert Answer

Bubich13
Answered 2022-01-05 Author has 4612 answers
First let’s look if the series converges absolutely.
For this, we need to see if \(\displaystyle\sum{b}_{{n}}=\sum{\frac{{{n}}}{{{2}^{{n}}}}}\) converges. And this is immediate using the ratio test
as \(\displaystyle\lim_{{{n}\to\infty}}{\frac{{{b}_{{{n}+{1}}}}}{{{b}_{{n}}}}}=\frac{{1}}{{2}}{ < }{1}\)
Conclusion: the given series converges absolutely hence converges
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Chanell Sanborn
Answered 2022-01-06 Author has 4782 answers
Hint. Your first thought is correct: look at the modulus.
Your reasoning about \(\displaystyle\infty\cdot{0}\) is wrong.
Try the ratio test.
If you know about the geometric series
\(\displaystyle{1}+{x}+{x}^{{2}}+\ldots\)
you can differentiate, multiply by x and actually find out what your series converges to.
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karton
Answered 2022-01-11 Author has 8659 answers

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

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