Infinite sum of non negative integers

[matrixone]

Homework Statement

Consider a sequence of non negative integers x₁,x₂,x₃,...x_n
which of the following cannot be true ?
$A)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}= \infty$

$B)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}< \infty$

$C)\sum ^{\infty }_{n=1} x_{n}< \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}< \infty$

$D)\sum ^{\infty }_{n=1} x_{n} \leq 5 \space and \space \sum ^{\infty }_{n=1} x_{n}^{2} \geq 25$

$E)\sum ^{\infty }_{n=1} x_{n}< \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}= \infty$

Homework Equations

The Attempt at a Solution

A) is true when x_n = n
B)
C) is true when x_n = 1/n
D)
E)

i can't find any ways to eliminate or finalise B,C, or D

 

[BvU]

Hi,
Let's work our way down the list. You did A already. Then:
If $x\ge1$, what do you know about $x^2$ in relation to $x$ ?

 

[PeroK]

The Attempt at a Solution

A) is true when x_n = n
B)
C) is true when x_n = 1/n
D)
E)

i can't find any ways to eliminate or finalise B,C, or D

It says the sequences are integers. $1/n$ is not an integer.

 

[BvU]

Hadn't even considered that ! was too focused on the fact that $C)\sum ^{\infty }_{n=1} x_{n}\nless \infty$ for 1/n

 

[matrixone]

Hi,
Let's work our way down the list. You did A already. Then:
If $x\ge1$, what do you know about $x^2$ in relation to $x$ ?

$x^2 \geq x$ and the equality is only when x=1
So in no case the sum of $x_{n}$ can exceed $x_{n}^{2}$
So B cannot be true .

Am i correct Sir ?

It says the sequences are integers. $1/n$ is not an integer.

I never noticed that SIr ! thanks for pointing out ...

if both the sequence contains only zeroes this is true ...
So C is also eliminated.

For D,

Lets have first sequence : 5,0,0,0,0,0,...
So second sequence : 25,0,0,0,0,0,...

So it is possible .

for E,
Only case where the first sum is less than infinity is finite number of positive terms. In that case second sum will also be finite.
So E is also true

So final answers B and E ?
Am i correct now ?
Thanks a lot both of you :)

 

[PeroK]

$x^2 \geq x$ and the equality is only when x=1
So in no case the sum of $x_{n}$ can exceed $x_{n}^{2}$
So B cannot be true .

Am i correct Sir ?
I never noticed that SIr ! thanks for pointing out ...

if both the sequence contains only zeroes this is true ...
So C is also eliminated.

For D,

Lets have first sequence : 5,0,0,0,0,0,...
So second sequence : 25,0,0,0,0,0,...

So it is possible .

for E,
Only case where the first sum is less than infinity is finite number of positive terms. In that case second sum will also be finite.
So E is also true

So final answers B and E ?
Am i correct now ?
Thanks a lot both of you :)

Looks like you've got it.