How do you use the integral test to determine if #ln2/sqrt2+ln3/sqrt3+ln4/sqrt4+ln5/sqrt5+ln6/sqrt6+...# is convergent or divergent?

Answer 1

The sum diverges.

The integral test for convergence says that if we have a series: #sum_(n=k)^oo f(n)#
Where #f# is positive, continuous and decreasing on the interval #[k,oo)#, the series converges if #int_k^oo f(x)\ dx# converges.
We can write our sum like this: #sum_(n=2)^oo ln(n)/sqrtn#
Unfortunately, our function isn't positive and decreasing on the interval #[2,oo)#, so we can't use the integral test just yet. We can however rewrite the sum like so: #sum_(n=2)^7 ln(n)/sqrtn+sum_(n=8)^oo ln(n)/sqrtn#
The function is positive and decreasing on #[8,oo)#, which means we can use the integral test on the right sum.
This means that the sum diverges if the integral #int_8^oo ln(x)/sqrtx\ dx# diverges.
To find the antiderivative, we begin by using integration by parts: #int\ f(x)g'(x)\ dx=f(x)g(x)-int\ f'(x)g(x)\ dx#
Letting #f(x)=ln(x)# and #g'(x)=1/sqrtx#, we get: #int\ ln(x)/sqrtx\ dx=2ln(x)sqrtx-int\ (2sqrtx)/x\ dx=#
#=2ln(x)sqrtx-2int\ x^(-1/2)\ dx=2ln(x)sqrtx-2*2sqrtx+C#
Now we can plug in the bounds of integration: #int_8^ooln(x)/sqrtx\ dx=[2ln(x)sqrtx-4sqrtx]_8^oo=#
#=lim_(x->oo)(2ln(x)sqrtx-4sqrtx)-(2ln(8)sqrt8-4sqrt8)=oo#

Since the integral diverges, the sum also diverges.

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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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