How do you use the integral test to determine the convergence or divergence of #Sigma 1/n^3# from #[1,oo)#?

Answer 1

The series:

#sum_(n=1)^oo 1/n^3#

is convergent.

Given the series:

#sum_(n=1)^oo 1/n^3#

we can choose as test function:

#f(x) = 1/x^3#
We have that #f(x)# is positive and decreasing in #(1,+oo)# that:
#lim_(x->oo) 1/x^3 = 0#
and finally that #f(n) = a_n#. Thus all the hypotheses of the integral test theorem are satisfied, and the convergence of the series is proved to be equivalent to the convergence of the improper integral:
#int_1^oo (dx)/x^3#

Calculating the integral we find:

#int_1^oo (dx)/x^3 = [-1/(2x^2) ]_1^oo #
#int_1^oo (dx)/x^3 = lim_(x->oo) (-1/(2x^2)) +1/2 = 1/2#

Thus the integral is convergent and then the series is proven to be convergent as well.

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Answer 2

To use the integral test to determine the convergence or divergence of the series Σ 1/n^3 from n = 1 to infinity, we need to compare it with the corresponding improper integral.

The integral test states that if f(x) is a continuous, positive, and decreasing function for x ≥ 1, and if the series Σ f(n) converges if and only if the integral ∫_1^∞ f(x) dx converges.

In this case, f(x) = 1/x^3. This function is continuous, positive, and decreasing for x ≥ 1.

Now, we evaluate the improper integral:

∫_1^∞ (1/x^3) dx = lim(a→∞) ∫_1^a (1/x^3) dx

= lim(a→∞) [-1/(2x^2)] from 1 to a

= lim(a→∞) [-1/(2a^2)] + 1/2

= 0 + 1/2

= 1/2

Since the integral ∫_1^∞ (1/x^3) dx converges (equals 1/2), by the integral test, the series Σ 1/n^3 from n = 1 to infinity also converges.

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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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