Summing a Series: Grouping Terms Legit?

[pivoxa15]

If one was to sum a series by grouping each two terms would that be legtimate?

i.e if you have an alternating series than grouping two terms would not be would it? i.e 1,-1,1,-1... would sum to 0 if you grouped them or 1 depending on how you group it.

Sometimes the only way to write a series under a summation sign is to group two elements in the series together.

 

[Office_Shredder]

For a finite series, you would be fine.

For an infinite series, not so much. The relevant things to look up are:

Convergent series
Absolutely convergent series

Basically, a convergent series has a limit of its partial sums (meaning you can't just skip by twos) that exists. If a series is absolutely convergent, the sum of the absolute values of the terms converges... and it turns out if it's absolutely convergent, you can re-arrange the terms (otherwise you can't).

 

[neutrino]

The sum of an infinite series, if it is convergent, is defined to be the limit of the partial sum S_n as n tends to infinity.

In this case, the partial sums take the form 1, 0, 1, 0...,1,0,...

 

[Gib Z]

They say the series does not converge because it oscillates.

In fact showing that groups terms in certain ways leads to different answers is a technique used to show some series are not absolutely convergent. The converse is not true though, showing certain groupings give the same answer does not mean it has absolute convergence, since there would be an infinite number of groupings possible.