Does Absolute Convergence Follow from Convergence in Series?

[oswald]

Firs Question

If the serie

∞
∑ Xn = (1/n²)
n=1

converge, then the serie |Xn| converge?

Second question

the serie
∞
∑ X^n / [fat(n)] diverge?
n=1

where fat(n) = n(n-1)(n-2)...4.3.2.1

Its important know the value of X in X^n?

 

[NateTG]

https://www.physicsforums.com/showthread.php?t=94383

 

[Marco_84]

Firs Question

If the serie

∞
∑ Xn = (1/n²)
n=1

converge, then the serie |Xn| converge?

Second question

the serie
∞
∑ X^n / [fat(n)] diverge?
n=1

where fat(n) = n(n-1)(n-2)...4.3.2.1

Its important know the value of X in X^n?

1)what is the || of 1/n^2?? think
2) search for geometric series and the hierarchy of infinities..

regards
marco

 

[HallsofIvy]

$$\sum_{n=0}^1 \frac{x^n}{n!}$$
is a well known Taylor's series for a simple function and your sum is only slightly different.

 

[oswald]

my answer
1)
∞
∑ Xn = (1/n²) = 1 + 1/4 + 1/9 ... its a p-serie and converge because p=2>1.
n=1
and
∞
∑ |Xn| = (1/n²) = 1 + 1/4 + ... its the same.. so its converge too.
n=1

2) converge using comparison test, no dought!

 

[Dick]

my answer
1)
∞
∑ Xn = (1/n²) = 1 + 1/4 + 1/9 ... its a p-serie and converge because p=2>1.
n=1
and
∞
∑ |Xn| = (1/n²) = 1 + 1/4 + ... its the same.. so its converge too.
n=1

2) converge using comparison test, no dought!

Better to use a ratio test. It will make it easy to see why the value of x doesn't matter.