Question

Determine whether the sequence converges or diverges. If it converges, find the limit. an = 9 + 14n2 n + 15n2 Step 1 To find lim n → [infinity] 9 + 14n2 n + 15n2 , divide the numerator and denominator by the highest power of n that occurs in the fraction. This is n .

Answer 1

Answer:

The sequence Converges

Step-by-step explanation:

Given the sequence $a_n = \frac{9+14n^{2} }{n+15n^{2} }$

To find the limit of the sequence, we will first divide the numerator and the denominator through by the highest power of n which is n² as shown;

$\lim_{n \to \infty} \frac{9/n^{2} +14n^{2}/n^{2} }{n/n^{2} +15n^{2}/n^{2} }\\ \lim_{n \to \infty} \frac{9/n^{2} +14 }{1/n +15n^{2}/n^{2 }}$

As $n$ tends to $\infty$, $\frac{a}{n}$ tends to zero where n is any constant, The limit of tyhe sequence as n tends to infinity becomes;

$= \frac{9/\infty+14 }{1/\infty+15 }\\= \frac{0+14}{0+15} \\= 14/15$

Therefore $\lim_{n \to \infty} \frac{9+14n^{2} }{n+15n^{2} } = 14/15$

Since the limit of the sequence gave a finite number , the sequence converges.

Note that the only case when the sequence diverges id when the limit of the sequence is infinite