I am being asked a question that I am not sure how to interpret: \"Write a defin

Question

I am being asked a question that I am not sure how to interpret:

"Write a definition of a sequence."

Would it be as simple as, a1, a2, a3, ... or would it be

I do not understand exactly what the professor is looking for. The rest of the question is:

"Give an example of a sequence converging to 7." And,

"Give an example of a diverging sequence."

All previous questions have just been straight off My Math Lab. This is followed by the same question applied to series. Any help would be appreciated.

''Write a definition of a sequence.'' Would it be as simple as, a1, a2, a3, ... or would it be n> N Rightarrow | an-L |

Explanation / Answer

Definition :

A sequence is usually defined as a function whose domain is a countable totally ordered set, although in many disciplines the domain is restricted, such as to the natural numbers. In real analysis a sequence is a function from a subset of the natural numbers to the real numbers.

---------------------------------------------------------------------------------------------------------------------------

Give an example of a sequence converging to 7

Solution :

Easiest to choose a geometric series that has a common ratio...

Now, we know that for such a sequence to converge, it must satisfy -1 < r < 1 , where r is the common ratio.

We know that the sum to infinity of a geometric sequence is :

S = first term / (1 - common ratio)

S = a / (1 - r)

Lets choose r = 6/7 because with this, we'd have denominator 1 - 6/7 ---> 1/7

This can converge to 7 when the first term, a = 1

So, choosing a = 1 , r = 6/7

S = a / (1 - r) --> 1 /(1 - 6/7) --> 1 / (1/7) --> 7 as needed

So, with a = 1 and r = 6/7, the sequence becomes :

1 , 6/7 , 6^2/7^2 , 6^3/7^3 , ..... upto infinity

1 , 6/7 , 36/49 , 216/343 , ....... ---> ANSWER for the sequence

-----------------------------------------------------------------------------------------------------

Give an example of a diverging sequence.

Solution :

This is quite easy because we can again choose a geometric sequence with common ratio lying outside the domain -1 < r < 1

So, lets choose a = 1 and r = 2 say

With this, sequence becomes :

1 , 2 , 2^2 , 2^3 , 2^4 , .... upto infinity

1 , 2 , 4 , 8 , 16 , ....... ----> ANSWER

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.