Consider the sequence a_n = ((-1)^n)/(sqrt(n))
What is the limit as n approaches infinity of a_n ?
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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:
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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