Rearranging the alternating harmonic series to sum to √2

[micromass]

Ah, yes. this is good.
Now, you've got that

$$\sum_{n=1}^{+\infty}{\frac{1}{n^2}}=\sum_{n=1}^{+\infty}{\frac{1}{(2n)^2}}+\sum_{n=1}^{+\infty}{\frac{1}{(2n+1)^2}}$$

You know two of the above series...

 

[andrey21]

ah yes i see simply solve to get:

1/4n^2 which equals pi^2/24 correct??

Then subtract to leave me value for 1/(2n+1)^2

 

[micromass]

Correct!

 

[andrey21]

Great thanks micromass :)

 

[andrey21]

I do have another series question micromass:
I have bin given the series:

5/4 + 1 + 4/5 + 16/25+...

It says describe what type of series this is?

Shall i try find general formula again?

 

[micromass]

The series is special type of serie. What special kinds of sequences/series have you seen?

Edited because I made a mistake somewhere (:

 

[andrey21]

Ye it says 5/4 at the beginning. does it look like it shouldn't be there??

 

[micromass]

Nono, the 5/4 is correct. I'm sorry about that confusion...

 

[andrey21]

Rite I am nt to sure what ur asking? special sequences??

 

[micromass]

Yes, you've probably seen about very special sequences/series.
For example, a very special kind of sequence are arithmetical sequences, these are sequences like 1,2,3,4,... or 2,4,6,8,... What other special sequences do you know?

 

[andrey21]

Ah arithmetical sequences are when u add or subtract a fixed value each time. eg:
0,3,6,9,...

 

[micromass]

Yes, are there other special sequences you know about?

 

[andrey21]

Erm well yes i do know many sequences am i looking for one in particular??

 

[micromass]

Well, you have arithmetic series and you have ... series.
Arithmetic series is where you add substract thesame value. What if you multiply/divide by thesame value?

 

[andrey21]

Ah i see is it going down by 0.8 each time:

1.25,1,0.8,0.64...

 

[andrey21]

is this correct?

 

[micromass]

Yes. so it is a geometric series...

 

[andrey21]

Rite so how would i express this:

sum a.(0.8)

How can i establish if it converges or not??

 

[micromass]

Have you not seen a special formula to determine the sum of a geometric sequence??
What do you know about geometric sequences?

 

[andrey21]

is it something like:

s = a/1-r

if so what values do each value represent?

 

[andrey21]

would it be s = 5/4 /(1-0.8)
= 6.25

correct?

 

[micromass]

Yes, that is OK!

 

[andrey21]

so it converges to 6.25 as n tends to infinity?

 

[micromass]

Yes, it will

 

[andrey21]

I have another series I have to describe and describe whether it converges micromass:
Its:

1+Pi/e + Pi^2/e^2+Pi^3/e^3

Could this be written as:

sum (Pi/e)^n

when n = 0

1

when n =1

Pi/e

etc...

 

[micromass]

Well, this series is another "special" series isn't it?

 

[andrey21]

Ye I've established its a geometrics series with ratio of Pi/e. and it diverges as r>1 correct?

 

[micromass]

correct!

 

[andrey21]

Great I do have one final problem I am stuck with. I have to use simple algebra to find the limit of:

Sqrt(n^2 + 3n) - n

I startd by multiplying top and bottom by Sqrt(n^2 + 3n) + n):

Which gives:

(Sqrt(n^2 + 3n) - n).(Sqrt(n^2 + 3n) + n)/Sqrt(n^2 + 3n) + n)

whcoh can be simplified to:

3n/Sqrt(n^2 + 3n) + n)

correct?? if so where do i go from here?

 

[micromass]

Factor out n in $$\sqrt{n^2+3n}+n$$. Then you can eliminate the n in the numerator...