# How do you determine the convergence or divergence of #Sigma ((-1)^(n+1)n)/(2n-1)# from #[1,oo)#?

The series:

is divergent.

You can determine whether an alternating series converges using Leibniz' criteria, which states that:

converges if:

As the general term of the series above can be expressed as:

We can quickly see that:

so that condition (ii) is not met and the series is divergent.

By signing up, you agree to our Terms of Service and Privacy Policy

To determine the convergence or divergence of the series ( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{2n-1} ), you can use the Alternating Series Test.

First, check if the terms of the series decrease in absolute value as ( n ) increases. In this series, the terms are ( \frac{n}{2n-1} ), which do decrease as ( n ) increases.

Next, check if the limit of the absolute value of the terms approaches zero as ( n ) approaches infinity. In this case, ( \lim_{n \to \infty} \left| \frac{n}{2n-1} \right| = \frac{1}{2} ), which is not zero.

Since both conditions of the Alternating Series Test are satisfied (terms decrease and limit approaches zero), the series ( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{2n-1} ) converges.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you use the Nth term test on the infinite series #sum_(n=1)^oorootn(2)# ?
- How do you Use an infinite geometric series to express a repeating decimal as a fraction?
- Why does the integral test not apply to #Sigma (2+sinn)/n# from #[1,oo)#?
- Show that #lim_( x->a) (x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3))=9/40 a^(-31/24)#?
- How do you find the positive values of p for which #Sigma n(1+n^2)^p# from #[2,oo)# converges?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7