Explaining Convergent Sequences w/ Examples

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A convergent sequence is one that approaches a specific value as the index increases. For example, the sequence defined by n+1=n/2+1/n, starting from n=1, converges to the square root of 2. Another example is the sequence {u_n}=\frac{3n^2-1}{n^2-5n}, which converges to 3. Many such convergent sequences exist, demonstrating the concept effectively. Understanding these examples helps clarify the definition of convergent sequences in mathematical analysis.
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Could anyone please explain "convergent sequence" with example.
 
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Consider this
n+1=n/2+1/n and start from n=1. Find n+1 continuously from this relation and you will find that this sequence converges to sqrt(2)

Also the sequence
<br /> {u_n}=\frac{3n^2-1}{n^2-5n}<br />

converges to 3.

One can find many convergent sequences like this.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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