Do infinite series always follow the associative property?

[zmike]

Intuitively I would think that [SUM of (-1)^n] or [SIGMA (-1)^n as n->infinity] Converges to 0 but apparently it doesn't according to the test for divergence ?

Does it actually converge or diverge? Why?

Thanks

 

[dx]

Hi zmike,

It doesn't converge:

1 = 1
1 + (-1) = 0
1 + (-1) + 1 = 1
1 + (-1) + 1 + (-1) = 0

It keeps oscillating between 1 and 0.

 

[zmike]

Hi zmike,

It doesn't converge:

1 = 1
1 + (-1) = 0
1 + (-1) + 1 = 1
1 + (-1) + 1 + (-1) = 0

It keeps oscillating between 1 and 0.

thanks, just one more thing,

so for ratio test, root test, test for divergence ALL work for alternating series? I am asking this b/c my textbook says they only work for positive sequences

also can I separate a sum? so can I take
[sigma an] * [sigma bn] = sigma (an*bn)? If not, wouldn't that violate the properties of limits?

Thanks

 

[dx]

The ratio test applies only to positive series.

For finite sums (∑a)(∑b) = ∑∑(ab) is true.

 

[zmike]

(∑a)(∑b) = ∑∑(ab)

What does the ∑∑ mean? take the sum twice? so is this also true with infinite sums?

also, does this mean that if either a or b is divergent, I can conclude that the series is divergent?

thanks

 

[dx]

Sorry about the confusing notation. Latex is not working currently and I didn't know how to write it.

Sum(i = 1 to n)[A_i] x Sum(j = 1 to m)[B_j] equals Sum(i = 1 to n)Sum(j = 1 to m)[A_i B_j]

For infinite series, if both the series are convergent, this will still work. If either of them is divergent, then clearly the left hand side is not defined.