How do you determine whether the sequence #a_n=(n+2)/n# converges, if so how do you find the limit?

Answer 1

It coverages to unity..

We have the sequence #{a_n}# where:

# a_n = (n+2)/n # # \ \ \ \ = 1+2/n #

And so as #n rarr oo => a_n rarr 1#

NB: A common result is that the harmonic series #sum_(r=1)^oo 1/r# diverges and hence the series #sum_(r=1)^oo a_r# diverges.

We can also visualise how the sequence behaves by looking at the graph of the function:

graph{(x+2)/x [-10, 10, -5, 5]}

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Answer 2

Emily Ammerman

To determine whether the sequence ( a_n = \frac{n+2}{n} ) converges, we can analyze its behavior as ( n ) approaches infinity. First, we simplify the expression by dividing both the numerator and denominator by ( n ), which gives us ( a_n = 1 + \frac{2}{n} ).

As ( n ) approaches infinity, the term ( \frac{2}{n} ) approaches zero because as the denominator becomes larger, the fraction becomes smaller. Therefore, the sequence ( a_n ) approaches the limit of ( 1 + 0 = 1 ) as ( n ) tends towards infinity.

In conclusion, the sequence ( a_n = \frac{n+2}{n} ) converges, and its limit is 1.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.