Why when finding the sum of a series does it go through m+1?

[burt]

[MATH]\sum\limits_{n=1}^ \infty a_n = a_1+a_2+a_3 + ... \Rightarrow\\ S_m = a_1+a_2+a_3+...+a_m+a_{m+1}[/MATH]

Why does [MATH]S_m[/MATH] go through [MATH]a_{m+1}[/MATH] and not [MATH]a_m[/MATH]? Where does the extra one come from?

 

[lev888]

[MATH]\sum\limits_{n=1}^ \infty a_n = a_1+a_2+a_3 + ... \Rightarrow\\ S_m = a_1+a_2+a_3+...+a_m+a_{m+1}[/MATH]

Why does [MATH]S_m[/MATH] go through [MATH]a_{m+1}[/MATH] and not [MATH]a_m[/MATH]? Where does the extra one come from?

It does seem strange but there is nothing that prevents you from defining S_m to be the sum of the first m+1 members. Please post more details from the source.

 

[JeffM]

Burt, this question has no context. You can define things however you want. If someone wants to define

[MATH]S_m = \sum_{i=1}^{m+1} a_i,[/MATH]
there is no contradiction, no ambiguity. So it is a legitimate definition. Whether it is useful or not depends on what the definer is trying to do. But you have not told us what the definer is trying to do.

 

[burt]

It does seem strange but there is nothing that prevents you from defining S_m to be the sum of the first m+1 members. Please post more details from the source.

So it is just a way of defining the sum - and you can do that however you want.

 

[JeffM]

So it is just a way of defining the sum - and you can do that however you want.

Exactly: it gives a compact symbol for a sum. But the notation is not intuitive. Context might make the choice plausible.

 

[pka]

[MATH]\sum\limits_{n=1}^ \infty a_n = a_1+a_2+a_3 + ... \Rightarrow\\ S_m = a_1+a_2+a_3+...+a_m+a_{m+1}[/MATH]

Why does [MATH]S_m[/MATH] go through [MATH]a_{m+1}[/MATH] and not [MATH]a_m[/MATH]? Where does the extra one come from?

I think it is most likely a typo. Many authors prefer that the index to start with zero as in:
\(\displaystyle {S_m} = \sum\limits_{k = 0}^m {{a_k}}\) So that there are indeed \(\displaystyle m+1\) terms in that sum.

 

[Dr.Peterson]

[MATH]\sum\limits_{n=1}^ \infty a_n = a_1+a_2+a_3 + ... \Rightarrow\\ S_m = a_1+a_2+a_3+...+a_m+a_{m+1}[/MATH]

Why does [MATH]S_m[/MATH] go through [MATH]a_{m+1}[/MATH] and not [MATH]a_m[/MATH]? Where does the extra one come from?

It will be very helpful if you quote the context in which this was stated. When you ask why someone else did something, you must either ask them what they were thinking, or look at what comes before (that might have suggested it) or what comes after (that might have been easier than otherwise because of the choice they made, looking ahead) to see if there is an explanation.

Without that context, we can only make guesses and tell you that it's not necessarily wrong. What makes it right (if anything) is a matter of context.

 

[Steven G]

I think it is most likely a typo. Many authors prefer that the index to start with zero as in:
\(\displaystyle {S_m} = \sum\limits_{k = 0}^m {{a_k}}\) So that there are indeed \(\displaystyle m+1\) terms in that sum.

Fine, but the last term would be a_m and not a_m+1

 

[pka]

Fine, but the last term would be a_m and not a_m+1

I said that I thought the \(\displaystyle a_{m+1}\) was a typo.

 

[Steven G]

I said that I thought the \(\displaystyle a_{m+1}\) was a typo.

OK, fine. Actually you did not say what the typo was, but all is fine.