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clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Evaluate the series?

Consider the series:
1/(2*3) + 1/(3*4) + ... + 1/[(k+1)(k+2)]

Which evaluates the series? Where is the flaw in the wrong one?

a) (1/2 - 1/3) + (1/3 - 1/4) + ... + [1/(k+1) - 1/(k+2)] + ... = 1/2
b) (1 - 5/6) + (5/6 - 3/4) + ... + [(k+3)/(2k+2) - (k+4)/(2k+4)] + ... = 1

clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Re: Evaluate the series?

Oh, note. I'm pretty sure that a) is correct and b) is not. But i'm unsure how to prove this. Especially for b.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Evaluate the series?

Both of the methods are correct, and can be verified by induction (although b. is a little tricky to prove).

The mistake is simply in the final evaluation of b.
The series is designed so that lots of terms cancel, and you're left with just 1 - (k+4)/(2k+4).

That right-hand term is equal to 1/2 + 1/k+2, and so as k --> ∞, it becomes equal to 1/2.
Therefore, method b) evaluates the series to be 1/2, agreeing with the other method.

(You can evaluate a. in the same way. Cancelling terms leaves you with 1/2 - 1/k+2, and this goes to 1/2.)

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

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Evaluate the series?

Hi all;

Another way:

Since it is so easy to telescope that series:

That takes care of a!

Now you only need to observe that:

Clearly b has been summed wrong, that is the error.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.