Answer to Question #4186 in Real Analysis for Junel

Question

Let (x sub n) := & 1/ln(n+1) for n element of natural numbers.&
1. use the definition of limit to show that lim(x sub n)=0&
2. find the specific value of k(e) as required in the definition of limit for each of & (i) e = 1/2 and & & (ii) e=1/10.

Answer 1

Task.Let (x sub n) := 1/ln(n+1) for n element of natural numbers.

1. use the definition of limit to show that lim(x sub n)=0
Proof.
We have to show that for any e>0 there exists and integer k(e)0
such that for all n > k(e) we have that
|x_n| < e
that is
|1/ln(n+1)| < e,
and so
(*) ln(n+1) > 1/e.
Since
lim_{x->+infinity} ln(x+1) = + infinity,
for any A>0 there exist an integer number M(A)>0 such that
(**) ln(m+1) > A for all m>M(A)

It remains to put A=1/e and k(e)=M(A).
Then (*) holds true.

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2. find the specific value of k(e) as required in the definition of limit for each of
(i) e = 1/2 and
(ii) e=1/10.
Solution.
We should find such k(e) that
ln(m+1) > 1/e
for all m>k(e), and so
m > exp(1/e)-1

Thus we can put
k(e) = [exp(1/e)],
where [t] means the integer part of t.
Now if e=1/2, then
k(1/2) = [exp(2)] = [7.38905609893065] = 7.
Now if e=1/10, then
k(1/10) = [exp(10)] = [22026.4657948067] = 22026