Having trouble with this infinite series and deciding whether it

burkinaval1b

burkinaval1b

Answered question

2022-01-04

Having trouble with this infinite series and deciding whether it converges or diverges.
The series:
n=1n(12i)n

Answer & Explanation

Bubich13

Bubich13

Beginner2022-01-05Added 36 answers

First let’s look if the series converges absolutely.
For this, we need to see if bn=n2n converges. And this is immediate using the ratio test
as limnbn+1bn=12<1
Conclusion: the given series converges absolutely hence converges
Chanell Sanborn

Chanell Sanborn

Beginner2022-01-06Added 41 answers

Hint. Your first thought is correct: look at the modulus.
Your reasoning about 0 is wrong.
Try the ratio test.
If you know about the geometric series
1+x+x2+
you can differentiate, multiply by x and actually find out what your series converges to.
karton

karton

Expert2022-01-11Added 613 answers

Consider n=0nzn. The radius of convergence is r=limn1n1n=1. Since |12i|=12, the series converges.

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