Show that the sum of the finite limits of these two series is also finite

In summary: If my question has caused any issues, I can only say that it would have been kind to reply to me with a quick question and I would have been happy to explain my doubt. That's all.
  • #1
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Homework Statement
Let Lim n>infinity of an and lim n>infinity of bn be real numbers, then show that their sum
lim n>infinity (an + bn) is also a real number. Hint, you can use the triangle inequality
Relevant Equations
Triangle inequality, Lim n>infinity(an), and Lim n>infinity(bn)
In the homework I am asked to proof this, the hint says that I can use the triangle inequality.
I was thinking that if both series go to a real number, a real number is just any number on the real number line, but how do I go from there,
 
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  • #2
rb120134 said:
Homework Statement:: Let Lim n>infinity of an and lim n>infinity of bn be real numbers, then show that their sum
lim n>infinity (an + bn) is also a real number. Hint, you can use the triangle inequality
Relevant Equations:: Triangle inequality, Lim n>infinity(an), and Lim n>infinity(bn)

In the homework I am asked to proof this, the hint says that I can use the triangle inequality.
I was thinking that if both series go to a real number, a real number is just any number on the real number line, but how do I go from there,
I suggest you need to take an arbitrary ##\epsilon > 0## to get started.
 
  • #3
PeroK said:
I suggest you need to take an arbitrary ##\epsilon > 0## to get started.
So let's say lim n> infity an =x then for every ε>0 there exists an N such that Ian-xI<ε for every n≥N. then if lim n> infity bn=y (where x and y are real numbers, given in question) for every epsilon greater then zero we can find an N such that I bn-yI<ε for every n≥N.

we have Ian-x +bn-yI<ε where ε>0 and n being equal or greater then N.

Now with the triangle inequality Ix+yI≤ IxI + IyI that implies
Ian-x +bn-yI≤ Ian-xI + I bn-yI so if we can make

I an-x I + I bn-yI <ε we know that Ian-x +bn-yI is also smaller then epsilon.

but I don't know how to conclude the proof, that this sum is also a real number
 
  • #4
rb120134 said:
but I don't know how to conclude the proof, that this sum is also a real number
One approach is to figure out precisely what real number it is!
 
  • #5
One technical point: these are sequences, not series. A series is a sum.
 
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  • #6
PeroK said:
One technical point: these are sequences, not series. A series is a sum.
yes you are right
 
  • #7
Notice you can find terms m, m' large-enough to make each difference less than ##\epsilon/2## instead of ##\epsilon##. And please try to learn Latex to make your posts more readable.
 
  • #8
rb120134 said:
So let's say lim n> infity an =x then for every ε>0 there exists an N such that Ian-xI<ε for every n≥N.

WWGD said:
And please try to learn Latex to make your posts more readable.

THAUROS said:
Wait, isn't using the reply button enough!?
What @WWGD is saying about LaTeX has nothing to do with the Reply button. Here is the first line I quoted redone using LaTeX:

So let's say ##\lim_{n \to \infty} a_n = x \Leftrightarrow ~\forall \epsilon > 0~ \exists N \ni ~|a_n - x| < \epsilon , \forall n \ge N##

You can click on what I wrote to see my LaTeX script for the above. There is also a LaTeX tutorial in the link at the lower left of the page.
 
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  • #9
Mark44 said:
What @WWGD is saying about LaTeX has nothing to do with the Reply button. Here is the first line I quoted redone using LaTeX:

So let's say ##\lim_{n \to \infty} a_n = x \Leftrightarrow ~\forall \epsilon > 0~ \exists N \ni ~|a_n - x| < \epsilon , \forall n \ge N##

You can click on what I wrote to see my LaTeX script for the above. There is also a LaTeX tutorial in the link at the lower left of the page.
Thanks. My question maybe wasn't clear enough. But since it nas cost me two warnings I have now contacted the platform moderators to please check and contact the user before issuing warnings.
Just to clarify, when I first read the thread I wasn't sure why the user was being asked to learn how to use Latex. What I meant was if it wasn't enough to copy, paste, and then click on the reply button.
If my question has caused any issues, I can only say that it would have been kind to reply to me with a quick question abd I would have been happy to explain my doubt. That's all.
Thanks again.
Best regards
 
  • #10
THAUROS said:
But since it nas cost me two warnings I have now contacted the platform moderators to please check and contact the user before issuing warnings.
Your profile doesn't show any warnings, so I'm not sure what you're talking about, unless it is the three posts of yours that were deleted. No warnings resulted from the post deletions.

THAUROS said:
Just to clarify, when I first read the thread I wasn't sure why the user was being asked to learn how to use Latex.
The comment from WWGD, and copied by me in post #8 was directed at the OP, because what he wrote was difficult to read.
THAUROS said:
What I meant was if it wasn't enough to copy, paste, and then click on the reply button.
This completely misses the point. Inequalities such as Ian-x +bn-yI≤ Ian-xI + I bn-yI (copied directly from one of the OP's posts) are much clearer when written using LaTeX. Compare the above to this:
$$|a_n - x + b_n - y| \le |a_n - x | + |b_n - y |$$
 
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  • #11
I don't think I missed the point at all. I just wasn't sure how to use Latex and I asked the correct question. Other than, thanks for explaining anyways.
 

1. What does it mean for a series to be finite?

A finite series is one in which the number of terms is limited and the sum of the terms is a finite number. This means that the series has a definite beginning and end, and the sum of all the terms in the series can be calculated.

2. Can you give an example of a finite series?

One example of a finite series is the arithmetic series: 1 + 2 + 3 + ... + n. This series has a finite number of terms, as it ends at n, and the sum of the terms can be calculated using the formula n(n+1)/2.

3. How do you prove that the sum of the finite limits of two series is also finite?

This can be proven using the properties of limits and series. First, we can take the limit of each series individually, which will give us a finite number. Then, we can add these two finite limits together to get the sum of the finite limits. Since both individual limits are finite, their sum must also be finite.

4. What is the significance of proving that the sum of the finite limits of two series is finite?

This proof is important in mathematics and science because it shows that the sum of two finite series is also finite. This means that we can use this property to manipulate and solve more complex series and equations.

5. Are there any exceptions to this rule?

Yes, there are some exceptions to this rule. For example, if one of the series has an infinite limit, then the sum of the finite limits of the two series will also be infinite. Additionally, if the two series have alternating signs, the sum of their finite limits may not be finite. However, in most cases, the sum of the finite limits of two series will be finite.

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