How do you use the Nth term test on the infinite series #sum_(n=1)^ooe^2/n^3# ?

Answer 1

Well, the nth term test (divergence test) does not give us any conclusion since the posted series is convergent.

By pulling #e^2# out of the summation, #sum_{n=1}^infty e^2/n^3=e^2sum_{n=1}^infty1/n^3#, which is a p-series with #p=3#. Since #p>1#, we can conclude that the series is convergent by P-Series Test.
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Answer 2

To use the Nth term test on the infinite series (\sum_{n=1}^{\infty} \frac{e^2}{n^3}), you need to examine the behavior of the individual terms of the series as (n) approaches infinity. Specifically, you need to check if the sequence of terms converges to zero as (n) approaches infinity.

The Nth term test states that if the terms of a series do not approach zero as (n) approaches infinity, then the series diverges. Conversely, if the terms do approach zero, it does not necessarily imply convergence; further tests may be required.

In this case, as (n) approaches infinity, (\frac{e^2}{n^3}) approaches zero, since the denominator grows much faster than the numerator. Therefore, according to the Nth term test, this series passes the test for convergence.

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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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