Home » Sequences » Page 2

Prove that a sequence cannot converge to two different limits

Prove that a sequence cannot converge to two different limits.

Proof. Consider a sequence such that

We show that we must have . By the definition of the limit we know that for all there exist a positive integers and such that

Let , then for all and all we have

Hence,

(The final line follows since if , then , so setting would contradict that for all .) Therefore, a sequence cannot converge to two different limits

Find an N such that the given convergent sequence is within ε of its limit

Consider the convergent sequence with terms defined by

Let . Find the value of and the value of such that for all for each of the following values of :

1. ,
2. ,
3. ,
4. ,
5. .

First, we know

since , so by (10.10) on page 380 of Apostol we know the limit is 0.
Then we have,

We reversed the inequality sign in the final step since since . Thus, if then for every we have . We compute for the given values of as follows:

1. implies .
2. implies .
3. implies .
4. implies .
5. implies .

Find an N such that the given convergent sequence is within ε of its limit

Consider the convergent sequence with terms defined by

Let . Find the value of and values of such that for all for each of the following values of :

1. ,
2. ,
3. ,
4. ,
5. .

First, we know

So then we have,

Thus, if then for every we have . We compute for the given values of as follows:

1. implies .
2. implies .
3. implies .
4. implies .
5. implies .

Find an N such that the convergent sequence is within ε of its limit

Consider the convergent sequence with terms defined by

Let . Find the value of and values of such that for all for each of the following values of :

1. ,
2. ,
3. ,
4. ,
5. .

First, we know

So then we have,

Thus, if then for every we have . We compute for the given values of as follows:

1. implies .
2. implies .
3. implies .
4. implies .
5. implies .

Find an N such that the convergent sequence is within ε of its limit

Consider the convergent sequence with terms defined by

Let . Find the value of and values of such that for all for each of the following values of :

1. ,
2. ,
3. ,
4. ,
5. .

First, we know

So then we have,

Thus, if then for every we have . We compute for the given values of as follows:

1. implies .
2. implies .
3. implies .
4. implies .
5. implies .

Find an N such that the convergent sequence is within ε of its limit

Consider the convergent sequence with terms defined by

Let . Find the value of and values of such that for all for each of the following values of :

1. ,
2. ,
3. ,
4. ,
5. .

First, we know

So then we have,

Thus, if then for every we have . We compute for the given values of as follows:

1. implies .
2. implies .
3. implies .
4. implies .
5. implies .

Determine the convergence or divergence of f(n) = ne-πin / 2

Consider the function defined by

Determine whether converges or diverges, and if it converges find its limit.

This sequence diverges.
Proof. We saw in this exercise (Section 10.4, Exercise #20) that the sequence defined by diverges. We could use this to show that this sequence diverges (since for we have and so cannot approach a finite limit). For practice, we can also prove this directly from the definition by contradiction as follows. Suppose there exists a real number and a positive integer such that for all and all we have

Since is positive we know and . So, letting , we have

Adding these two expressions and using the triangle inequality we have,

This is a contradiction. Hence, the sequence does not tend to a limit .

Determine the convergence or divergence of f(n) = (1 / n) e-πin / 2

Consider the function defined by

Determine the convergence or divergence of the sequence and if it converges determine its limit.

This sequence converges since

But each of the limits in the product exists since

and

Hence,

Therefore, (as we saw in this exercise, if is complex valued and then the sequence as well). Hence, the sequence converges to 0.

Determine the convergence or divergence of f(n) = e-πin / 2

Consider the function defined by

Determine whether the sequence converges or diverges, and if it converges determine its limit.

The sequence diverges.
Proof. First, we use DeMoivre’s theorem to write,

(Note: On page 380 Apostol claims (without proof) that the sequence defined by is divergent. If you want to accept that then this sequence will diverge since a complex-valued sequence diverges if and only if the sequences defined by the functions and converge. Since it is a good exercise (and maybe Apostol wanted us to prove it ourselves) we can prove this is divergent from the definition.)

Suppose, for the sake of contradiction, that the sequence converges to a limit . Then we know for every there exists a positive integer such that for all we have

Since is positive we know and . Hence, taking we have,

Adding these two expressions and using the triangle inequality,

a contradiction. Hence, the sequence diverges

Determine the convergence or divergence of f(n) = (1 + i/2)-n

Consider the function defined by

Determine whether the sequence converges or diverges, and if it converges find the limit.

First, we note that

Then,

Therefore (since and is real) we know from (10.10) on page 380 of Apostol that

Hence, there exists an such that for all we have

So , so the sequence converges to 0.