The proof of convergence. I am confused with the summation

[flyingpig]

Homework Statement

[tex]Prove\; that\;if\;\sum_{n=1}^{\infty} a_n \;converges,\;then \lim_{n\to\infty}a_n = 0[/tex]

Book solution

[tex]s_n= a_1 + a_2 +...+a_n[/tex]

[tex]s_{n-1}= a_1 + a_2 +...+a_{n-1}[/tex]

[tex]a_n=s_n-s_{n-1}[/tex]

Then they did a few limits, and proved that the difference is 0. BUt that is not my question.

My question is this part
[tex]s_{n-1}= a_1 + a_2 +...+a_{n-1}[/tex]

If it is n - 1, why are they starting from a₁? Shouldn't it be a_0/sub]?

 

[gb7nash]

No. If we're assuming that the first term in the sum is [tex]a_{1}[/tex], the (n-1)th partial sum is defined to be [tex]s_{n-1}= a_1 + a_2 +...+a_{n-1}[/tex], i.e. it's the sum of the first term, the second term, ... , and the (n-1)th term.

 

[flyingpig]

Yeah exactly so it should be a₀

 

[gb7nash]

In your problem, the first term will always be [tex]a_{1}[/tex]. The partial sum that we choose won't affect the first term. We could have [tex]s_{n}[/tex], [tex]s_{n-1}[/tex], [tex]s_{n+3}[/tex], but in each case the first term will always be [tex]a_{1}[/tex]

If you're still not convinced, take a look at:

http://mathworld.wolfram.com/PartialSum.html

However, if the series defined is given as [tex]\sum_{n=0}^{\infty} a_n[/tex], then you would be right. The first term of all partial sums would start at [tex]a_{0}[/tex]. It just depends on the first term that's defined.