How do you use the direct comparison test to determine if #sume^(-n^2)# from #[0,oo)# is convergent or divergent?

Answer 1

Converges by Direct Comparison Test.

We have

#sum_(n=0)^ooe^(-n^2)=sum_(n=0)1/e^(n^2)#
We see #a_n=1/e^(n^2)#
For the comparison sequence, we'll use #b_n=1/e^n=(1/e)^n>=a_n# for all #n# on #[0, oo),# as we have a smaller denominator (due to removing the squared #n#) and therefore a larger sequence.
We know #sum_(n=0)^oo(1/e)^n# converges, as it's a geometric series with the absolute value of the common ratio #|r|=1/e<1#.
Thus, since the larger series converges, so does the smaller series #sum_(n=0)^ooe^(-n^2)# by the Direct Comparison Test.
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Answer 2

To use the direct comparison test to determine the convergence or divergence of the series ( \sum e^{-n^2} ) from ( n = 0 ) to infinity, we can compare it to another series whose convergence or divergence is known.

Since ( e^{-n^2} ) is always positive, we can compare it to ( \frac{1}{n^2} ), which is also always positive.

We know that ( e^{-n^2} < \frac{1}{n^2} ) for all ( n ) greater than or equal to 1, because ( e^{-n^2} ) decreases more rapidly than ( \frac{1}{n^2} ).

Since the series ( \sum \frac{1}{n^2} ) is convergent (it's a p-series with ( p = 2 > 1 )), and ( e^{-n^2} < \frac{1}{n^2} ) for all ( n \geq 1 ), by the direct comparison test, ( \sum e^{-n^2} ) is also convergent.

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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.

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.

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.

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.

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