christian0710 said:
I almost got it all now!
There is only ONE thing left that confuses me. It's the idea of taking the limit of a partial sum (Not the Sequence of a partial sum which makes more sense) which is what they do in my book.
They're taking the limit of a
sequence, the sequence of partial sums.
I think I see where you are confused -- between the general form of the partial sum and a particular partial sum (for a specific value of n).
Maybe this will make it clearer. For the series, ##\sum_{i = 1}^{\infty}\frac{1}{2^n}##, the general term in the sequence of partial sums is ##\frac{2^n - 1}{2^n}##.
The second term in the sequence of partial sums is 3/4 (when n = 2). Since 3/4 is a constant, its limit as n gets large is just 3/4. For any specified value of n, the corresponding term in the sequence of partial sums is a constant; hence its limit is just that same number.
However, the general term in the sequence is ##\frac{2^n - 1}{2^n}##, and
its limit as n gets large, is 1.
christian0710 said:
The partial sum does not go to infinity according to the definition:
Sn=Ʃai=a1+a2+a3+...+an.
Try to separate the ideas of n going to infinity versus the limit of the partial sum going to infinity as n goes to infinity. The three examples I had in my previous post showed sequences that had different behaviors as n "went to infinity."
christian0710 said:
For me it makes sense to take the limit of a sequence of partial sums where the terms go to infinity (just like taking the limit of a function where x goes to infinity)
...where n goes to infinity.
When we're talking about sequences (which are functions), the argument of the function is usually n, a
discrete variable that takes integer values. A variable such as x is usually used to denote a
continuous variable that can take on all of the real values in an interval.
christian0710 said:
In my book under the phrase "The sum of the series is the limit of the sequence of the partial sums" they show
Ʃan=Lim Ʃai where Ʃai = Sn. How can they take a limit of a sum/series that only goes to n?
You're omitting information about the index of the summation, so are obscuring some important information.
S
n = a
1 + a
2 + ... + a
n = ## \sum_{i = 1}^n a_i##.
The sum of the series (if such a sum exists) is
$$ \lim_{n \to \infty} S_n = \lim_{n \to \infty} \sum_{i = 1}^n a_i$$
S
n is a function of n. As n gets larger, S
n represents a (finite) sum with more terms.
christian0710 said:
So i understand that we take the limit of a sequence Lim an= 1/2n we get 0
the sum of this series is 1, do we also say "the limit of the series" Lim Ʃ1/2n"?
Remember what I said a lot earlier in this thread about a series being associated with
two sequences.
For an arbitrary series ## \sum_{i = 1}^{\infty} a_i##, the two sequences are:
1. The sequence of terms in the series: {a
1, a
2, ..., a
n, ...}
2. The sequence of partial sums: {S
1, S
2, ..., S
n, ...}
For the series we have been discussing, where a
i = 1/2
i, we have
1. ## \lim_{i \to \infty} a_i = \lim_{i \to \infty} 1/2^i = 0##, and
2. ## \lim_{i \to \infty} S_i = \lim_{i \to \infty} \frac{2^i - 1}{2^i} = 1##
The limit that is most important is the second one - the limit of the sequence of partial sums. The first limit above, is of much less importance. For a series to converge, it is
necessary for the limit of the terms in the series to be zero, but this is not
sufficient.
For example, in the well-known harmonic series, ## \sum_{n= 1}^{\infty} \frac{1}{n}##,
## \lim_{n \to \infty} \frac{1}{n} = 0##, but it turns out that the series itself diverges. From this we can conclude that the limit of the sequence of partial sums must diverge.
christian0710 said:
This is the last part I'm having trouble understanding: taking limits af sequences (makes sense) vs. taking limits of sums/series (makes little sense to me)
You don't take the limit of a series (an infinite sum), but you can take the limit of a finite sum, where the number of terms in the sum depends on an index n (IOW, is a function of n).
[/quote]
christian0710 said:
I just can't grasp this; the limit 1/2n is 0 (makes sence), but the limit of Ʃ1/2n is what? "the limit of a sum" how Am i to imagine that?
See above.
christian0710 said:
Edit: I just added a picture the sum and limit of sum as I understand it, could this be the right way of understanding it?
The sum you show in the picture is a finite sum: 1 + 2 + 3 + ... + n.
The formula n(n + 1)/2 is the sum of the first n integers. It does not represent the sum of an infinite number of integers.
Here's a question you asked in the thing you attached:
And the limit of the formula n(n + 1)/2 is 1?
Does the sequence {1, 3, 6, 10, ...} look like it is approaching 1?
