How do you determine the convergence or divergence of #Sigma ((-1)^(n)n)/(n^2+1)# from #[1,oo)#?
the series is convergent.
An alternative series:
converges if the succession
So we make the test:
As the denominator is always positive we can focus on the numerator:
which is always true, for
The we check that:
So both conditions are satisified and the series is convergent.
Another way that showing
Examining the sign of the derivative, we see that the denominator is always positive. Thus the sign of the derivative as a whole is dependent on the sign of the numerator. When
A negative derivative shows a decreasing function. Thus,
Using this in conjunction with the fact that
we can claim that
We can go on to note that