Question

How is infinite sum different to limiting sum? Or are they the same things?

Answer 1

Okay, thank you.

Answer 2

I'm double checking your question. Just getting the definitions right now

Answer 3

Infinite Sum = an infinite sequence of numbers that is added (sum)
Literally infinite numbers
A sequence of numbers is like 1, 2, 3, 4, 5, 6, 7 and so on
What do you feel an infinite sequence would be like? How many numbers?
The answer is in the name
Infinite
That means endless

Answer 4

Okay, thank you. How about limting sum?
It's not endless?

Answer 5

@bedpotato

An Infinte Sum, in a generic sense, is defined to be a sum, like @Tutor.Stacey mentioned, is an infinite sequence of numbers added. This definition however, does not imply the behaviour of the infinite sum, i.e. does it converge/diverge?

This is where a Limitting Sum comes in to play. A limitting sum is an infinite sum that converges to a certain value, i.e it doesn't get infinitely large, instead it approaches a certain value as you add more and more terms.

Answer 6

Okay, so if a question asked for e.g. "for what values of x does this series NOT have an infinite sum" Does that mean it doesnt have a limiting sum or it does?

Answer 7

Give me a moment, while I work on your question.

Answer 8

what math class is this question for please?

Answer 9

Well, in Australia, it's year 12. Series and sequences.

Answer 10

oh, I see.

Answer 11

Here is an example of an infinite sum:\[\bf \sum_{x=1}^{\infty}x=1+2+3+4+5....\]But a Limitting Sum, i.e. a sum that actually approaches a certain value as we add more and more terms would be like this:\[\bf \sum_{x=1}^{\infty} \left( \frac{ 1 }{ 2 } \right)^{x-1}=1+\frac{ 1 }{ 2 }+\frac{ 1 }{ 4 }...\]Notice that they're both infinite sums. However the limitting sum is such that each successive term gets much smaller than the previous term(s) and so the sum actually converges to a "limit" instead of getting infinitely. More formally, a sum is such that:\[\bf \lim_{n \rightarrow \infty}\sum_{x=1}^{n}a_x=\lim_{n \rightarrow \infty}(a_1+a_2+...+a_n)=I, | \ I \in \mathbb{R}\]This just means that a limitting sum is an infinite sum which has a "limit", i.e. that it converges to some value 'I'.

Answer 12

@bedpotato

Answer 13

Okay, thankyou. I understand the difference, but for the question I gave above, how would you go about solving it? Sorry for so many questions.

Answer 14

the answer to your previous question is it does not have a limiting sum.
an infinite sum means that the sum has no limit. It has infinite possibilities.
Does that make sense? That is what infinite means. :)

Answer 15

so |r| > 1 ? as opposed to |r| < 1 ?

Answer 16

No no, |r| < 1. For any geometric series to be convergent, it must satisfy \(\bf |r|<1\).

Answer 17

yes, -Genius- Is correct!

Answer 18

Sorry, I'm just bit busy answering other people's questions.

Answer 19

@bedpotato A geometric series is one given in the form:\[\bf \sum_{x=1}^{n}ar^{x-1}\], Where 'a' is the first term and 'r' is the common ratio which must satisfy |r| < 1. Also note that in my example for a limitting sum above, it is also a geometric series and it converges to a certain value, i.e. has a limit, because \(\bf |r|=\frac{1}{2}<1\). Hence it's a limitting sum since it converges.

Answer 20

For limiting sum it would be |r| < 1, but the question is asking for values for not an infinite sum? Let's say the series if 1, (x-1)^2, (x-2)^4...
r = (x-1)^2. If the questin said: for what values of x does this series not have in inifine sum? We have to find the values for which |r| > 1.

Answer 21

I mean the series is: 1, (x-1)^2, (x-1)^4. Sorry. I appreciate your help.

Answer 22

@bedpotato 'r' must be a constant...it's the common ratio for the geometric series. Also PLEASE note that the restriction |r| < 1 only implies the convergence of a geometric series given in the form \(\bf \sum ar^{n-1} \), where 'a' and 'r' are constants. It does not imply the convergence of any other infinite series.

Answer 23

@bedpotato If you have a specific question from homework, then please post the whole thing and I'll try and answer it.

Answer 24

You were correct about the limiting sum. ok . so now you are looking for values of x, correct?

Answer 25

@genius12 Okay, I undertand. Thank you. And that is the whole question.
@Tutor.Stacey Yes, I am.

Answer 26

x would need to be any number less than or equal to 2.
sorry less than 2.
Your value for each number in the series needs to be less than 1.
therefore the value of x-1 must be less than 0
Does that make sense?