Define the notation used here in describing a series?

[whitejac]

Homework Statement

"For the given series, write formulas for the sequences a_n , S_n, R_n and find the limit as n->∞ (if it exists)

Homework Equations

∑^∞₁ ((1/n) - 1/(n+1)

The Attempt at a Solution

I know how to take the limit, that's no problem. I'm a bit confused about what a_n , S_n, R_n are referring to. Maybe I'm just misunderstanding the usage...

My book speaks generally of defining them (by that i mean, no formal definition is given but it often refers to a previous example) however it does say this:

Let us call the terms of the series a so that the series
is
"a1 + a2 + a3 + a4 + · · · + a_nn + · · · ."
Remember that the three dots mean that there is never a last term; the series goes
on without end. Now consider the sums Sn that we obtain by adding more and
more terms of the series. We define
S1 = a1,
S2 = a1 + a2,
S3 = a1 + a2 + a3,
· · ·
Sn = a1 + a2 + a3 + · · · + an

It then goes further to say:

The difference Rn = S − Sn is called the remainder (or the remainder after n
terms). From (4.6), we see that...

Where S is the lim n-> ∞. So can Rn not be found if there's no limit?
Also, why even write down a_n if it's just the same as S_n?

 

[andrewkirk]

So can $R_n$ not be found if there's no limit?

That is correct.

why even write down $a_n$ if it's just the same as $S_n$?

They are not the same. $S_n$ is the sum of $a_1$ to $a_n$, ie
$$S_n\triangleq \sum_{i=1}^n a_i$$
Also, in words sometimes a distinction is made by using the word 'sequence' to refer to the $a_n$s and 'series' to refer to the $S_n$s. It's good to bear that in mind to head off potential confusion if they use that terminology.

 

[Ray Vickson]

Homework Statement

"For the given series, write formulas for the sequences a_n , S_n, R_n and find the limit as n->∞ (if it exists)

Homework Equations

∑^∞₁ ((1/n) - 1/(n+1)

The Attempt at a Solution

I know how to take the limit, that's no problem. I'm a bit confused about what a_n , S_n, R_n are referring to. Maybe I'm just misunderstanding the usage...

My book speaks generally of defining them (by that i mean, no formal definition is given but it often refers to a previous example) however it does say this:

Let us call the terms of the series a so that the series
is
"a1 + a2 + a3 + a4 + · · · + a_nn + · · · ."
Remember that the three dots mean that there is never a last term; the series goes
on without end. Now consider the sums Sn that we obtain by adding more and
more terms of the series. We define
S1 = a1,
S2 = a1 + a2,
S3 = a1 + a2 + a3,
· · ·
Sn = a1 + a2 + a3 + · · · + an

It then goes further to say:

The difference Rn = S − Sn is called the remainder (or the remainder after n
terms). From (4.6), we see that...

Where S is the lim n-> ∞. So can Rn not be found if there's no limit?
Also, why even write down a_n if it's just the same as S_n?

Who says that $a_n$ is the same as $S_n$? YOU wrote
$$S_n = a_1 + a_2 + \cdots + a_n$$
$S_n$ and $a_n$ look a lot different to me.

Anyway, if $S = \lim_{n \to \infty} S_n$ then it makes sense to speak of the difference between $S$ and $S_n$; that difference is called $R_n$. In other words, $S = S_n + R_n$.

In some (rare) cases we can say exactly what is $R_n$ (for example, for a geometric series), but usually we have to be content with producing bounds on the value of $R_n$ instead of an exact value. Still, that is often very useful, especially when we want to evaluate an infinite series numerically by stopping at $S_n$ for some finite $n$. A bound on $R_n$ will tell us an upper bound on the error we would be making by stopping at $n$.

 

[Stephen Tashi]

You need to explain what "the given series" is.

From the information you quoted, apparently $a_n$ is the nth term of a series.

Is the "given series" the series whose $n$-th term is $a_n = \frac{1}{n} - \frac{1}{n+1}$ ?$S_n$ is the "$n$-th partial sum" of the series. $S_n = \sum_{i=1}^n a_i$, so $S_n$ is not the same thing as $a_n$.$S$ is the sum of the series $S = \lim_{n \rightarrow \infty} S_n$

$R_n$ is $S - S_n$ You are correct that $R$ is undefined if $S$ doesn't exist.

 

[whitejac]

You need to explain what "the given series" is.

From the information you quoted, apparently $a_n$ is the nth term of a series.

Is the "given series" the series whose $n$-th term is $a_n = \frac{1}{n} - \frac{1}{n+1}$ ?$S_n$ is the "$n$-th partial sum" of the series. $S_n = \sum_{i=1}^n a_i$, so $S_n$ is not the same thing as $a_n$.$S$ is the sum of the series $S = \lim_{n \rightarrow \infty} S_n$

$R_n$ is $S - S_n$ You are correct that $R$ is undefined if $S$ doesn't exist.

How can you say that a_n is a term if the question requests a formula?
and I apologize, the given series is the series found in section two under "useful formulas,"

∑∞1 ((1/n) - 1/(n+1)

This is the series with which I am supposed to be determining the formulas "an, Sn, Rn."
If a_n is the nth term in the series, then that would just be the given equation, right?
if n = 27, then the nth term would be
((1/27) - (1/28))
Then Sn would be the partial sum to the nth term, which would be
(1/1) - (1/2) + (1/2) - (1/3) + (1/3) - 1/4 ... + (1/27) - (1/28) = 1 - (1/28) = (27/28)
am i understanding the definitions correctly?

 

[LCKurtz]

How can you say that a_n is a term if the question requests a formula?
and I apologize, the given series is the series found in section two under "useful formulas,"

This is the series with which I am supposed to be determining the formulas "an, Sn, Rn."
If a_n is the nth term in the series, then that would just be the given equation, right?
if n = 27, then the nth term would be
((1/27) - (1/28))
Then Sn would be the partial sum to the nth term, which would be
(1/1) - (1/2) + (1/2) - (1/3) + (1/3) - 1/4 ... + (1/27) - (1/28) = 1 - (1/28) = (27/28)
am i understanding the definitions correctly?

Almost. That last line is the sum to the $27$th term $S_{27}$, not to the $n$th term $S_n$.

 

[Mark44]

How can you say that a_n is a term if the question requests a formula?

write formulas for the sequences a_n , S_n, R_n

In other words, from the given summation (which is NOT an equation), write expressions for each of the sequences listed above.

and I apologize, the given series is the series found in section two under "useful formulas,"

This is the series with which I am supposed to be determining the formulas "an, Sn, Rn."
If a_n is the nth term in the series, then that would just be the given equation, right?

No, and again, the summation you showed is not an equation. An equation has this symbol in it = .

if n = 27, then the nth term would be
((1/27) - (1/28))
Then Sn would be the partial sum to the nth term, which would be
(1/1) - (1/2) + (1/2) - (1/3) + (1/3) - 1/4 ... + (1/27) - (1/28) = 1 - (1/28) = (27/28)
am i understanding the definitions correctly?

When you're working with infinite series, such as $\sum_{j = 1}^{\infty} \frac 1 j$, there are several bits of terminology that beginning students have trouble with.

A sequence is basically a list of numbers. A series is the sum of a possibly infinite number of terms of the related sequence of terms.

With an infinite series, you have the following:

• There is the underlying sequence of terms, $\{a_1, a_2, a_3, \dots \}$, which in my example is the sequence {1/1, 1/2, 1/3, 1/4, ...}
• There is the sequence of partial sums, $\{S_1, S_2, S_3, \dots \}$, which in my example is the sequence {1, 3/2, 11/6, ...}. S₁ is the sum of the first term in the sequence of terms (IOW, it's the first term). S₂ is the sum of the first two terms in the sequence of terms, S₃ is the sum of the first three terms, and so on.
• There is the series itself. A series converges if and only if its sequence of partial sums converges.

 

[Stephen Tashi]

How can you say that a_n is a term if the question requests a formula?

I don't see any difficulty in giving a formula as the answer for a "term". The n-th term is given by a function of n.

If a_n is the nth term in the series, then that would just be the given equation, right?

There is no given "equation". (There is no "=" sign in the given expression $\sum_{1}^\infty (\frac{1}{n} - \frac{1}{n+1} )$ .) The given information is "an expression" showing a summation that involves a function of n. Yes, $a_n$ is the given function.

Most text would have written the given sum as $\sum_{n=1}^\infty (\frac{1}{n} - \frac{1}{n+1} )$ Did your text leave out the "n = 1" ?

if n = 27, then the nth term would be
((1/27) - (1/28))

Yes.

Then Sn would be the partial sum to the nth term, which would be
(1/1) - (1/2) + (1/2) - (1/3) + (1/3) - 1/4 ... + (1/27) - (1/28) = 1 - (1/28) = (27/28)

Yes, with the understanding that n = 27.

am i understanding the definitions correctly?

I think so.

 

[whitejac]

Hmm, I think I understand now. The definitions and nomenclature definitely slips me up. Thank you all for your patience!