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A bounded sequence need not be convergent

Can you show me an illustration which shows a sequence that is convergent?

I don't understand when if lim n ---> infinity s_{n}= l, the sequence s_{n}converges to l or {s_{n}}. Now what is l ?

My attempt or understanding of the convergence of sequence or series is similar to that of curve tracing. which I think is wrong. please tell me why are we performing root's test or ratio test and other test on the sequence? If it's to find whether or not the series is convergent then please explain what convergence is. Please tell me how curve tracing is different from convergence and divergence. I am getting confused between curve tracing and convergence.

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# When a sequence is convergent

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