# How do you find the sum of #2/1+4/3+8/9+16/27+...+2^(n+1)/3^n+...#?

To find the sum of ( \frac{2}{1} + \frac{4}{3} + \frac{8}{9} + \frac{16}{27} + \ldots + \frac{2^{n+1}}{3^n} + \ldots ), we can recognize that each term is a geometric series. The common ratio between consecutive terms is ( \frac{2}{3} ). Using the formula for the sum of an infinite geometric series, which is ( \frac{a}{1 - r} ), where ( a ) is the first term and ( r ) is the common ratio, we can calculate the sum. So, the sum of the given series is ( \frac{\frac{2}{1}}{1 - \frac{2}{3}} ). Simplifying this expression yields ( \frac{6}{1 - \frac{2}{3}} ), which further simplifies to ( \frac{6}{\frac{1}{3}} ). Thus, the sum of the series is ( 18 ).

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

so:

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 show the convergence of the series #(n!)/(n^n)# from n=1 to infinity??
- How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma (cos(npi))/(n+1)# from #[1,oo)#?
- How do you use the Integral test on the infinite series #sum_(n=1)^oo1/root5(n)# ?
- How do you determine whether #1/(n!)# convergence or divergence with direct comparison test?
- How do you determine whether the infinite sequence #a_n=(2n)/(n+1)# converges or diverges?

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