Why is the infinite series 1 - 1 + 1 - 1 + 1 - 1... considered divergent?

[eprparadox]

Hello!

How can I justify that the infinite series 1 - 1 + 1 - 1 + 1 - 1... is divergent?

If I were to look at this, I see every two terms canceling out and thus, and assume that it is convergent since the sum doesn't blow up. That's what my intuition would tell me.

I know I can use different tests to figure out that it is divergent, but I don't have an intuition for why it's so.

Any ideas? Thanks!

 

[micromass]

By definition, a series $\sum a_n$ is convergent if the sequence

a_1,~a_1+a_2,~a_1+a_2+a_3,~a_1+a_2+a_3+a_4,...

is convergent.

So in your case, you have to investigate the sequence

1,~1-1,~1-1+1,~1-1+1-1,~1-1+1-1+1,...

Is this sequence convergent?

 

[eprparadox]

So you're saying that for a series to converge, it's the series of partial sums (that's the correct term right?) must also converge?

And it's just alternating between the values 0 and 1 infinitely. So yeah it is divergent.

Thanks so much.

 

[micromass]

So you're saying that for a series to converge, it's the series of partial sums (that's the correct term right?) must also converge?

Yes, the sequence of partial sums must converge. That's the definition of when a series converges.

 

[Mark44]

For a given series $\sum_{n = 0}^{\infty}a_n$, there are two sequences that are involved:
The sequence of terms in the series: {a₀, a₁, a₂, ... , a_n, ...}.
The sequence of partial sums: {a₀, a₀ + a₁, a₀ + a₁ + a₂, ... }.

As micromass already said, if the sequence of partial sums converges to a number, then the series itself converves to that same number.

Note that I showed a series that starts with an index of 0. The starting index can be some other integer.

 

[eprparadox]

awesome, thanks so much guys!