# Does #a_n=(n + (n/2))^(1/n) # converge?

The sequence is convergent.

so

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

To determine if the sequence (a_n = \left(n + \frac{n}{2}\right)^{\frac{1}{n}}) converges, we need to analyze its behavior as (n) approaches infinity.

First, let's rewrite (a_n) using properties of exponents:

[a_n = \left(n \left(1 + \frac{1}{2}\right)\right)^{\frac{1}{n}} = \left(\frac{3}{2}n\right)^{\frac{1}{n}}]

Now, we can use the limit definition of (e) (the base of the natural logarithm) to evaluate the limit:

[\lim_{n \to \infty} \left(\frac{3}{2}n\right)^{\frac{1}{n}} = e^{\lim_{n \to \infty} \frac{\ln\left(\frac{3}{2}n\right)}{n}}]

By applying L'Hôpital's Rule to the indeterminate form (\frac{\infty}{\infty}), we get:

[= e^{\lim_{n \to \infty} \frac{\frac{d}{dn}\left(\ln\left(\frac{3}{2}n\right)\right)}{\frac{d}{dn}n}} = e^{\lim_{n \to \infty} \frac{\frac{\frac{3}{2}}{\frac{3}{2}n}}{1}}]

[= e^{\lim_{n \to \infty} \frac{1}{n}}]

Since the denominator (n) grows without bound as (n) approaches infinity, the limit of (\frac{1}{n}) approaches zero.

Thus, the limit of (a_n) simplifies to (e^0 = 1).

Therefore, the sequence (a_n = \left(n + \frac{n}{2}\right)^{\frac{1}{n}}) converges, and its limit is 1.

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 test the improper integral #int (2x-1)^-3dx# from #[0,1/2]# and evaluate if possible?
- How do you find the sum of finite geometric series?
- How do you test the alternating series #Sigma (-1)^n(2^n+1)/(3^n-2)# from n is #[0,oo)# for convergence?
- How do you find the positive values of p for which #Sigma (1/n(lnn)^p)# from #[2,oo)# converges?
- How do you find #lim sin(2x)/ln(x+1)# as #x->0# using l'Hospital's Rule?

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