Infinite sum of non negative integers

1. Mar 7, 2017

matrixone

1. The problem statement, all variables and given/known data
Consider a sequence of non negative integers x1,x2,x3,...xn
which of the following cannot be true ?
$A)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}= \infty$

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

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

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

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

2. Relevant equations

3. The attempt at a solution

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

i cant find any ways to eliminate or finalise B,C, or D

2. Mar 7, 2017

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 $x$ ?

3. Mar 7, 2017

PeroK

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

4. Mar 7, 2017

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

5. Mar 7, 2017

matrixone

$x^2 \geq x$ 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 :)

6. Mar 7, 2017

PeroK

Looks like you've got it.