Consider the sequence a_n = ((-1)^n)/(sqrt(n))

What is the limit as n approaches infinity of a_n  ?

 

Expert Answers

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First consider `b_n = 1/sqrt(n)`

As `n->oo`, `sqrt(n) -> oo`, so `1/sqrt(n)->0`

(Think of dividing 1 pizza between more and more people.  Each person gets a smaller and smaller piece.)

Now consider `a_n = (-1)^n/sqrt(n)`

The `(-1)^n` means the terms alternate positive and negative.  To see this, consider the first few terms:

`-1/sqrt(1),1/sqrt(2),-1/sqrt(3),1/sqrt(4),...,1/sqrt(1000000),...`


These numbers are still getting close to 0, they just bounce back and forth from positive to negative, along the way.

Thus `lim_(n->oo) a_n = 0`

 

 

 

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