How do you determine whether the sequence #a_n=2^n-n^2# converges, if so how do you find the limit?
Using the binomial expansion we have that:
Or:
As:
we have:
So:
and then:
By signing up, you agree to our Terms of Service and Privacy Policy
To determine convergence of the sequence (a_n = 2^n - n^2), consider the behavior of (a_n) as (n) approaches infinity. Note that (2^n) grows exponentially while (n^2) grows polynomially. As (n) becomes large, the exponential term (2^n) dominates the polynomial term (n^2). Therefore, the sequence diverges to positive infinity as (n) approaches infinity.
Hence, the sequence (a_n = 2^n - n^2) diverges, and there is no limit.
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.
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.
- Determine whether the series # sum_(n=1)^oo (2n^2 +3n)/sqrt(5+n^5)# is convergent or divergent. How do i tell which comparison test to use?
- How do you determine the convergence or divergence of #Sigma ((-1)^n n!)/(1*3*5***(2n-1)# from #[1,oo)#?
- How do you use the ratio test to test the convergence of the series #∑ 3^n/(4n³+5)# from n=1 to infinity?
- How do you test the improper integral #int x^(-1/3) dx# from #(-oo, oo)# and evaluate if possible?
- How do you determine if #a_n=(1+n)^(1/n)# converge and find the limits when they exist?

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