Convergence of sequence: Limit of a Sequence & Solved Examples

Ex-1: Consider a sequence 2, 3/2 , 4/3 , 5/4, …….. here Sn = 1 + 1/n

A1. From the given sequence we can see that the sequence Sn is convergent.

\(\lim_{n \rightarrow\ \infty}S_{n}\)

=\(\lim_{n \rightarrow\ \infty}(1+\frac{1}{n})\)

=\(1+\frac{1}{\infty}\)

=\(1+0\)

=\(1\)

Since the limit of the sequence is 1, therefore the sequence Sn is convergent .

Ex-2: Consider a sequence Sn= n² + (-1)ⁿ.

A2. From the given sequence we can see that the sequence sn is divergent.

\(\lim_{n \rightarrow\ }S_{n} \)

=\(\lim_{n \rightarrow\ \infty} n^{2} + (-1)^{n}\)

=\(\infty^{2} + (-1)^{\infty}\)

=\(\infty\)

Since the limit of the sequence is \(\infty\), therefore the sequence Sn is diververgent .

Ex-3.Consider a sequence \9Sn=\frac{n^{2}+2}{n^{2}+5}\).

A2. From the given sequence we can see that the sequence sn is divergent.

\(\lim_{n \rightarrow\ \infty}S_{n}\)

=\(\lim_{n \rightarrow\ \infty}\frac{n^{2}+2}{n^{2}+5} \)

=\(\lim_{n \rightarrow\ \infty}\frac{n^{2}(1+\frac{2}{n^{2}})}{n^{2}(1+\frac{5}{n^{2}})} \)

=\(\lim_{n \rightarrow\ \infty}\frac{(1+\frac{2}{n^{2}})}{(1+\frac{5}{n^{2}})} \)

= \(\frac{1}{1})

=\(1\).

Since the limit of the sequence is 1, therefore the sequence Sn is convergent .

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