IDNLearn.com connects you with a community of experts ready to answer your questions. Find reliable solutions to your questions quickly and accurately with help from our dedicated community of experts.

Determine the limit of the sequence or state that the sequence diverges.

[tex]\[ a_n = 6 - \frac{3}{n^2} \][/tex]

(Use symbolic notation and fractions where needed. Enter DNE if the sequence diverges.)

[tex]\[ \lim_{n \rightarrow \infty} a_n = ? \][/tex]


Sagot :

Let's determine the limit of the sequence [tex]\( a_n = 6 - \frac{3}{n^2} \)[/tex] as [tex]\( n \)[/tex] approaches infinity.

1. We start by examining the given sequence:
[tex]\[ a_n = 6 - \frac{3}{n^2} \][/tex]

2. To find the limit as [tex]\( n \)[/tex] approaches infinity, we analyze each term separately:

- The term [tex]\( 6 \)[/tex] remains constant as [tex]\( n \)[/tex] approaches infinity.

- The term [tex]\( \frac{3}{n^2} \)[/tex] changes as [tex]\( n \)[/tex] grows larger. Since [tex]\( n^2 \)[/tex] becomes very large as [tex]\( n \)[/tex] approaches infinity, [tex]\( \frac{3}{n^2} \)[/tex] becomes very small.

3. Formally, as [tex]\( n \)[/tex] approaches infinity, [tex]\( n^2 \rightarrow \infty \)[/tex].

4. Hence, [tex]\( \frac{3}{n^2} \rightarrow 0 \)[/tex].

5. Therefore, the sequence [tex]\( 6 - \frac{3}{n^2} \)[/tex] approaches:
[tex]\[ 6 - 0 = 6. \][/tex]

6. Thus, we conclude that the limit of the sequence [tex]\( a_n \)[/tex] as [tex]\( n \)[/tex] approaches infinity is:
[tex]\[ \lim_{n \rightarrow \infty} a_n = 6 \][/tex]

So, the limit of [tex]\( a_n \)[/tex] is [tex]\(\boxed{6}\)[/tex]. The initial value given in the problem ([tex]\( \lim_{n \rightarrow \infty} a_n = 1 \)[/tex]) appears to be incorrect based on the analysis.