How do you know if the series #sum 1/(n^(1+1/n))# converges or diverges for (n=1 , ∞) ?
2 Answers
is divergent.
Explanation:
Using the properties of exponents:
Note now that:
As:
and the exponential function
Consider now the harmonic series
that we know to be divergent.
Using the limit comparison test:
we can see that, as the limit of the ratio is finite, the two series have the same character and also:
is divergent.
The series diverges
Explanation:
To test the convergence of the series
We need to calculate the limit
Now,
According to the limit comparison test , since this limit is a finite nonzero number, the series
However, it is well known that