How do you find the interval of convergence of #Sigma (x+10)^n/(lnn)# from #n=[2,oo)#?

Answer 1

The series:

#sum_(n=0)^oo (x+10)^n/lnn#

is convergent for # x in [-11,9)# and absolutely convergent in the interior of the interval.

(i) For every #n > 2# we have #ln n < n#, so if # ( x+10 ) >= 1#:
# (x+10)^n/ln n > 1/n#

and the series is divergent.

(ii) For #abs(x+10) < 1# we have:
#abs(x+10)^n/ln n < abs(x+10)^n#
Since #sum_(n=0)^oo abs(x+10)^n# is a geometric series of ratio #r < 1#, it is convergent and then also:
#sum_(n=0)^oo (x+10)^n/lnn# is absolutely convergent.
(iii) For #(x+10) <= -1 # we can write the series as an alternating series:
#sum_(n=0)^oo (-1)^n abs(x+10)^n/lnn#

and apply the Leibniz test:

Clearly for #(x+10) < -1# we have
#lim_(n->oo) a_n = lim_(n->oo) abs(x+10)^n/lnn = oo #
and the series is not convergent, while for #(x+10) = -1#
#lim_(n->oo) a_n = lim_(n->oo) 1/lnn = 0 #

and

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

so the series is convergent.

In conclusion the series is convergent for:

#-1 <= x+10 < 1#
That is for # x in [-11,9)#
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Answer 2

To find the interval of convergence of the series Σ((x+10)^n / ln(n)) from n=2 to infinity, we can use the ratio test. According to the ratio test, the series converges if the absolute value of the ratio of successive terms approaches a finite number as n approaches infinity.

So, we calculate the limit as n approaches infinity of |((x+10)^(n+1) / ln(n+1)) / ((x+10)^n / ln(n))|.

This simplifies to the absolute value of ((x+10)(n) / (n+1)) * (ln(n) / ln(n+1)).

As n approaches infinity, ln(n) / ln(n+1) approaches 1, and (n / (n+1)) approaches 1.

Thus, the limit simplifies to |x + 10|.

For the series to converge, |x + 10| must be less than 1.

Therefore, the interval of convergence is -11 < x < -9.

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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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