How do you apply the ratio test to determine if #Sigma 1/(lnn)^n# from #n=[2,oo)# is convergent to divergent?

Answer 1

he series:

#sum_(n=2)^oo 1/(lnn)^n#

is convergent.

We have the series:

#sum_(n=2)^oo 1/(lnn)^n#

Now evaluate the ratio:

#abs(a_(n+1)/a_n) = abs ( (1/(ln(n+1)^(n+1)) ) /(1/(lnn)^n)) = (lnn)^n/(ln(n+1)^(n+1)) = (lnn/ln(n+1))^n 1/ln(n+1)#

Now consider the function:

#f(x) = lnx/ln(x+1)#
the limit for #x->oo# is in the form #oo/oo# so we can calculate it using l'Hospital's rule:
#lim_(x->oo) lnx/ln(x+1) = lim_(x->oo) (d/dx lnx)/(d/dx ln(x+1)) = lim_(x->oo) (1/x)/(1/(x+1)) = lim_(x->oo) (x+1)/x =1 #
As #f(n) = lnn/ln(n+1)# we then have:
#lim_(n->oo) lnn/ln(n+1) =1#

and therefore:

#lim_(n->oo) (lnn/ln(n+1))^n =1#

Then:

#lim_(n->oo) abs(a_(n+1)/a_n) = lim_(n->oo) (lnn/ln(n+1))^n 1/ln(n+1) = lim_(n->oo) 1/ln(n+1) = 0#

which proves the series to be convergent.

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Answer 2

To apply the ratio test to determine the convergence or divergence of the series Σ 1/(ln n)^n from n=2 to infinity, you calculate the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term. If the limit is less than 1, the series is convergent. If the limit is greater than 1 or does not exist, the series is divergent.

The (n+1)th term of the series is 1/(ln(n+1))^(n+1). Therefore, the ratio of the (n+1)th term to the nth term is [1/(ln(n+1))^(n+1)] / [1/(ln n)^n]. Simplifying this gives [(ln n)^n] / [(ln(n+1))^(n+1)]. Taking the limit as n approaches infinity of this ratio will determine the convergence or divergence of the series.

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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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