SOLUTION: Determine if the following series are convergent of divergent. If the series is geometric find the sum as well: [(-1)^n +1] / sqr n (NOTE: +1 is separated from ^n) at n =1 to

Algebra ->  Sequences-and-series -> SOLUTION: Determine if the following series are convergent of divergent. If the series is geometric find the sum as well: [(-1)^n +1] / sqr n (NOTE: +1 is separated from ^n) at n =1 to       Log On


   

Question 579908: Determine if the following series are convergent of divergent. If the series is geometric find the sum as well:
[(-1)^n +1] / sqr n (NOTE: +1 is separated from ^n)
at n =1 to infinity

Thank you so much!
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
It is not a geometric series.
For odd n, (-1)^n=-1 and [(-1)^n+1]=0, so a%5Bn%5D=0
For even n, (-1)^n=1 and [(-1)^n+1]=2, so a%5Bn%5D=2%2Fsqrt%28n%29 .
In general, 0%3C=+a%5Bn%5D%3C=2%2Fsqrt%28n%29 , so a%5Bn%5D can be made less than any epsilon by just making n%3E4%2Fepsilon%5E2. So it meets the definition of converging to zero.
However, the way you are expected to prove that it converges may be different.