# ?

Aakriti Sharma
14 Points
12 years ago

ans- .

HARIKRISHNAN M
18 Points
12 years ago

FIRST CALCULATE THE Tnth TERM

IT IS

n/((2n+1)!)

just sum it out to find answer

sigma(n/(2n+1)!)

sigma(.5*2n/(2n+1)!)

i.e

.5*sigma((2n+1-1)/(2n+1)!)

.5*sigma(1/(2n-1)!+1/(2n+1)!)

.5*(1/1!-1/3!+1/3!-1/5!..........-1/(2n+1)!)

cancelling we get

0.5*(1-1/(2n+1)!)

the question is not toughest as you say

Sathya
35 Points
12 years ago

For me,its tough!Im a year younger than u.....Anyways thnks for ur ans

HARIKRISHNAN M
18 Points
12 years ago

we both are of same age .

Sathya
35 Points
12 years ago

And also,ur ans is wrong

290 Points
12 years ago

Hi Satyaram,

That is because, nobody gets the r-th term of the series right in this.

The r-th teerm in the series is

Tr = r(r!)2r/(2r+1)!.

And this is not quite easy to sum. You try to sum this.

Dont ask for the solution. It is a good summation question to do on your own.

Regards,

Sathya
35 Points
12 years ago

Dude,

The difficult thing is to find the rth term...

How did it strike to u that Tr = r(r!)2r/(2r+1)! when u saw the question?

290 Points
12 years ago

Dude,

When you look at some particular term, say for example the 4th term which is

4/(1.3.5.7.9)

the obvious thing is the Nr is "r"

now comes the tricky part, the Dr (the number of terms in the Dr varies, and are products of odd numbers)

So you have 1.3.5.7.9 - now to represent this in mathematicl notation you need to have (1.2.3.4.5.6.7.8.9)

So 1.3.5.7.9 = (1.2.3.4.5.6.7.8.9)/(2.4.6.8) = (9!)/244!

so for the r-th term the Dr is (2r+1)!/2rr!

And hence you get the Tr.

Regards,

290 Points
12 years ago

But then that is not the toughest part.

The tougher part is to sum it. :-)

Regards,

Sathya
35 Points
12 years ago

Dude,

I got an more easier solution dude :

Multiply and divide by 2.

Now,

Write the numerator as difference of the first and last term.i.e.

Write

and

and so on....

Now take the 1/2 common and:::

u get

which is

HARIKRISHNAN M
18 Points
12 years ago

Hey this is what was my answer.

I admit my answer is wrong but what is wrong with my nth term

290 Points
12 years ago

Harikrishnan,

The n-th term is not

n/(2n+1)!. For eg put n=2, your expression will give 2/5! = 2/(1.2.3.4.5). But in the question the 2nd term is 2/(1.3.5).

Ok Satyaram,

For of all there is no solution to your question anywhere above, for you to give a simpler solution :-)

And more over the solution has not been typed!!!.... it has been copy-pasted :-)

That solution is the only way to solve it.

Nice that you got it.

Regards,

Sathya
35 Points
12 years ago

Dude,

Macha,I got the answer from another friend.

This is latex dude :) and he finished it in easier way...

And Ive seen this type of question,actually its an trignometric series,in which after simplyifying it...they use this method,this method is common for all the series question i think so :)

And Of course,i copy-pasted it .... (Ennaala Eppidi yosikamudiyum boss.........)

And also I want to know ur solution ..........

And also see artofproblemsolving.com ,be an member and its also an common forum for all maths olympiads all over the world,MIT students r the experts of this site (Like IItians in askiitian ),u can post,ask and clear ur doubts :)

290 Points
12 years ago

Dude,

That is the only solution, as I had stated previously.

Do the same thing with the r-th term [ie the general term of the series, so that it will be over with just one term]

If you note

Tr = r(r!)2r/(2r+1)!

= 1/2 [ (r-1)!2r-1/(2r-1)! - r!2r/(2r+1)! ]

= 1/2 [ tr-1 - tr ]

So, the summation upto infinite terms = 1/2

And summation upto n-terms is 1/2 [ 1 - n!2n/(2n+1)! ]

Regards,

HARIKRISHNAN M
18 Points
12 years ago

omg i am sorry by (2n+1)! i mean you have to substitue 1 then 2 and so on to n and not substituting n and calulating the factorial.

ok now it is okkk is nt it

or if confusion persists

just replace (2n+1)!(at all places where you see it) with ∏(2n+1) where ∏ denotes product just like sigma giving n values from 1 to n

then the final answer would be

.5(1-1/(∏(2n+1)))

or

.5(1-(2^n)*n!/(2n+1)!)

hope you all are clear with it

sorry for any misconceptions

290 Points
12 years ago

Harikrishnan,

Exactly. Now the change is fine.