Sequence Convergence: Determine if an= (13(4n)+11)/(10(5n))

• nvalia
Since the limit of (4/5)^n is 0 and the limit of (1/5)^n is also 0, we can say that the sequence an converges to 0. In summary, the sequence an converges to 0.
nvalia

Homework Statement

Determine if the sequence an converges.

an = (13(4n)+11)/(10(5n))

N/A

The Attempt at a Solution

$$\lim_{n \to \infty} (\frac{(3(4^n))+11}{10(5^n)})$$
= $$1/10 \lim_{n \to \infty} (\frac{(3(4^n))+11}{5^n})$$
= $$1/10 \lim_{n \to \infty} (\frac{(3(4^n)}{5^{n}}) + \lim_{n \to \infty} (\frac{11}{5^n})$$

I feel like I am missing something very basic. Thank you for your help!

Last edited:
nvalia said:

Homework Statement

Determine if the sequence an converges.

an = (13(4n)+11)/(10(5n))

N/A

The Attempt at a Solution

$$\lim_{n \to \infty} (\frac{(3(4^n))+11}{10(5^n)})$$
= $$1/10 \lim_{n \to \infty} (\frac{(3(4^n))+11}{5^n})$$
= $$1/10 \lim_{n \to \infty} (\frac{52^{n}}{5^{n}}) + \lim_{n \to \infty} (\frac{11}{5^n})$$

I feel like I am missing something very basic. Thank you for your help!
how did you get your last line? where did 52 come from?

what is $$\lim_{n \to \infty} (\frac{4}{5})^n$$?

Oops! That was a typo -- fixed. Sorry about that.

And, if I write out the sequence, I would have to say the limit of (4/5)^n = 0.
Am I allowed to split the limit like I did, and, if so, are the limits separately 0 and 0, so that the final answer is "converges to zero"?

it depends what has been proved, but where the limie exists and is finite for both f & g, generally it is fine to assume
$$\lim_{n \to \infty} (f(n) +g(n)) = \lim_{n \to \infty} f(n) +\lim_{n \to \infty}g(n)[/$$

$$\lim_{n \to \infty} (\frac{(3(4^n))+11}{10(5^n)}) = \lim_{n \to \infty} \frac{3}{10}(\frac{4}{5})^n + \lim_{n \to \infty} \frac{11}{10}(\frac{1}{5})^n$$

1. What is sequence convergence?

Sequence convergence refers to the behavior of a sequence as it approaches a specific limit or value. In other words, it is the process of determining whether a sequence tends towards a particular value as its terms increase.

2. How is sequence convergence determined?

Sequence convergence is determined by calculating the limit of the sequence as n approaches infinity. If the limit exists and is finite, the sequence is said to be convergent. If the limit does not exist or is infinite, the sequence is divergent.

3. What is the formula for determining the limit of a sequence?

The formula for determining the limit of a sequence is: lim(n->∞) an = L, where "an" represents the sequence and "L" represents the limit as n approaches infinity.

4. How is the given sequence an= (13(4n)+11)/(10(5n)) analyzed for convergence?

To analyze the given sequence for convergence, we can first simplify the expression by dividing both the numerator and denominator by 5n. This results in an= (52n+11)/(50n). Then, we can divide the coefficients of n to get an= (52/50 + 11/50n). As n approaches infinity, the term 11/50n becomes smaller and smaller, approaching 0. Therefore, the limit of the sequence is 52/50, or 1.04. This means that the sequence is convergent.

5. What does it mean if a sequence is divergent?

If a sequence is divergent, it means that its terms do not approach a specific limit as n approaches infinity. This could mean that the terms of the sequence are increasing or decreasing without bound, or that they oscillate between different values without approaching a specific limit.

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