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alexdok [17]
4 years ago
15

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 .
Mathematics
1 answer:
konstantin123 [22]4 years ago
3 0

Answer:

<h2>The sequence Converges</h2>

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.

<u>Note that the only case when the sequence diverges id when the limit of the sequence is infinite</u>

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