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#1 2007-02-12 05:40:41

woodoo
Member
Registered: 2007-02-10
Posts: 11

Help with a series

I have this problem that I can't figure out.
                                                                                                                                                                         
a_n is a sequence of positive #s. For each n in the natural #s, b_n = (a1+a2+...+an)/n. And I have to use this to show


∑ b_n   
n=1     
                                                                                                                                                                   
diverges to positive infinity. Any1 have any ideas?

Last edited by woodoo (2007-02-12 05:42:16)

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#2 2007-02-16 02:48:52

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,956

Re: Help with a series

Let me try:-
I am not using any standard test, just logic.
b_1=1/1=1
b_2=3/2=1.5
b_3=6/3=2
b_4=10/4=2.5
b_5=15/5=3
b_6=21/6=3.5
b_7=28/7=4
The sum of n natural numbers is given by the formula n(n+1)/2.
When this is divided by n, (n+1)/2 is obtained.

∑b_n is the same as
n=1


∑ (n+1)/2.
n=1

As n->∞, the sum is also ∞, therefore the series is
Divergent!

Have I made a mistake somewhere? I guess not smile


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#3 2007-02-16 03:33:09

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Help with a series

While you're right for that sequence, I think you need to be more general. Instead of choosing a_1, a_2, a_3 to be 1, 3, 6 etc., I think it's required to show that it's true for any set of numbers.


Why did the vector cross the road?
It wanted to be normal.

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